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FIRST   LESSONS 


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THOMAS  HTIL. 


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J-Vj^JTS    BfP'^"''    REASONING 


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Digitized  by  the  Internet  Arciiive 

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FIEST  LESSONS 


GEOMETRY. 


BY 


THOMAS  HILL. 


FACTS   BEFORE   REASONING. 


BOSTON: 

UICKLING,    SWAN,    AND    BROWN. 

1855. 


"^^^ 


V 


GIFTO0' 


Entered  according  to  Act  of  Congress,  in  the  year  1S55,  by 
THOMAS    HILL, 

In  the  Clerk's  Office  of  the  District  Court  of  the  District  of  Massachusetts. 


S  tereotyped  by 
HOBART    &    ROBBINS, 

NEW  ENGLAND  TYPE  AND  8TERE0TTPE  F0UUDK7, 
B  0  S  T  Q  N  . 


PREFACE. 


I  HAVE  long  been  seeking  a  Geometry  for  beginners,  suited  to 
my  taste,  and  to  my  convictions  of  what  is  a  proper  foundation 
for  scientific  education.  Finding  that  Mr.  Josiaii  Holbrook 
agreed  most  cordially  with  me,  in  my  estimate  of  this  study,  I 
had  hoped  that  his  treatise  would  satisfy  me  ;  but,  although 
the  best  I  had  seen,  it  did  not  meet  my  views.  Meanwhile, 
my  own  children  w^ere  in  most  urgent  need  of  a  text-book,  and 
the  sense  of  their  want  has  driven  me  to  take  the  time  neces- 
sary for  writing  these  pages.  Two  children,  one  of  five,  the 
other  of  seven  and  a  half,  were  before  my  mind's  eye  all  the 
time  of  my  writing  ;  and  it  will  he  found  that  children  of  this 
age  are  quicker  at  comprehending  first  lessons  in  Geometry 
than  those  of  fifteen.  Many  parts  of  this  book  will,  however, 
be  found  adapted,  not  only  to  children,  but  to  pupils  of  adult 
age.  The  truths  are  sublime.  I  have  tried  to  present  them  in 
a  simple  and  attractive  dress. 

I  have  addressed  the  child's  imagination,  rather  than  his 
reason,  because  I  wished  to  teach  him  J;q  conceive  of  forms. 


8S2^16v' 


IV  PREFACE. 

The  child's  powers  of  sensation  are  developed  before  his  powers 
of  conception,  and  these  before  his  reasoning  powers.  This  is, 
therefore,  the  true  order  of  education  ;  and  a  powerful  logi- 
cal drill,  like  Colburn's  admirable  first  lessons  of  Arithmetic, 
is  sadly  out  of  place  in  the  hands  of  a  child  whose  powers  of 
observation  and  conception  have,  as  yet,  received  no  training 
whatever.  I  have,  therefore,  avoided  reasoning,  and  simply 
given  interesting  geometrical  facts,  fitted,  I  hope,  to  arouse  a 
child  to  the  observation  of  phenomena,  and  to  the  perception 
of  forms  as  real  entities. 

In  the  pronunciation  of  words  at  the  foot  of  the  page  the 
notation  of  Dr.  Worcester's  Dictionaries  has  been  followed. 

Waltham,  Mass.,  JVbv.  1854. 


CONTENTS. 


CHAPTER.  PAGS 

I.  What  is  this  Book  about? 7 

II.  Points,  Lines,  and  Planes, 9 

III.  About  Straight  Lines  and  Curves,  ....  11 

IV.  About  Angles, 13 

V.  Parallel  Lines, 16 

VI.  About  Triangles, 19 

VII.  More  about  Triangles, 21 

Vni.  Right  Triangles, 24 

IX.  Similarity  and  Isoperimetry,      ...*..  26 

X.  The  Size  of  Triangles, 30 

XI.  Different  Kinds  of  Triangles, 33 

XII.  Quadrangles, 36 

XIII.  Parallelograms, 40 

XIV.  Rectangles  and  Squaresj 43 

XV.  Triangles  and  Rectangles, 46 

XVI.  Circles, 52 

XVII.  More  about  Circles,     55 

XVni.  Measuring  Angles, 59 

1* 


VI  CONTENTS. 

XIX.  Chords, 62 

XX.  Chords  and  Tangents, 66 

XXI.  More  about  Chords  and  Tangents,    .   .    .    G9 

XXII.  Inscribed  Polygons, 73 

XXIII.  More  about  Inscribed  Polygons,     ....    77 

XXIV.  How   MUCH    further    round  a  Hoop  than^ 

across  it? 81 

XXV.  How  to  Measure  Circles  and  their  Parts,    85 

XXVI.  About  Curvature, 89 

XXVII.  About  a  Wheel  Rolling, 92 

XXVIII.  More  about  the  Rolling  Wheel,    ....    95 

XXIX.  Wheels  Rolling  round  a  Wheel,    ....    98 

XXX.  Wheel  Rolling  in  a  Hoop, 101 

XXXI.  Hanging  Chain, 105 

XXXII.  Path  of  a  Stone  in  the  Air, 109 

XXXIII.  The  Shadow  of  a  Ball, 112 

XXXIV.  The  Shadow  of  a  Reel, ...115 

XXXV.  The  Cow's  Foot  in  a  Cup  of  Milk,    ...  120 

XXXVI.  Cubes 125 

XXXVII.  About  Cones,     129 

XXXVin.  The  Sphere, 132 


CHAPTER   I. 

WHAT   THIS   BOOK   IS   ABOUT. 

1.  I  HAVE  written  a  little  book  for  you  about 
Geometry.  You  will  find  a  great  many  new 
words  in  it;  but  I  have  taken  pains  to  explain 
them  all,  and  I  think  you  will  understand  them 
all,  if  you  will  only  begin  at  the  beginning, 
and  read  each  chapter  very  carefully  before  you 
go  to  another.  And  if  you  find  any  place  in  the 
book  that  you  cannot  understand,  I  think  you  will 
do  well  to  turn  back,  and  read  the  whole  over 
again,  from  the  second  chapter.  When  you  come 
again  to  the  place  which  you  did  not  understand 
before,  I  think  you  will  find  it  has  grown  easier 
for  you. 

2.  I  hope  you  will  find  the  book  interesting.  It 
tells  about  straight  lines,  and  circles,  and  many 
different  curves,  and  a  few  solid  bodies.  It  will 
tell  you  curious  things  about  the  shadows  of  mar- 
bles, and  the  rolling  of  hoops,  and  about  tossing  a 
ball,  and  other  plays  for  children.     But,  if  some 


8  INTRODUCTION. 

parts  of  the  book  do  not  seem  interesting,  you 
should  study  those  parts  all  the  more  carefully ; 
for  they  maj,.  perliaps,  be  the  most  useful  parts. 

3.  Geometry  is  the  most  useful  of  all  the  sci- 
'ir.cGs."  To'  undt3J:3tand  Geometry,  will  be  a  great 
help  in  learning  all  the  other  sciences;  and  no 
other  science  can  be  learned  unless  you  know 
something  of  Geometry.  To  study  it,  will  make 
your  eye  quicker  in  seeing  things,  and  your  hand 
steadier  in  doing  things.  You  can  draw  better, 
write  better;  cut  out  clothes,  make  boots  and 
shoes,  work  at  any  mechanical  trade,  or  learn  any 
art,  the  better  for  understanding  Geometry.  And, 
if  you  w^ant  to  understand  about  plants  and  ani- 
mals, and  the  wonderful  way  in  which  the  All- wise 
Creator  has  made  them,  you  must  learn  a  little 
Geometry,  for  that  explains  the  shapes  of  all 
things. 

4.  In  this  book  I  can  teach  you  but  little.  I 
hope,  however,  that  it  will  be  enough  to  make  you 
want  to  knoYf  more.  I  shall  tell  you  only  the 
easiest  and  most  interesting  things  now  ;  but  when 
you  are  older  you  may  study  what  is  more  diffi- 
cult. '  Many  of  the  things  that  I  shall  tell  you 
will  be  very  curious,  and  you  will,  perhaps,  vfonder 
how  men  can  find  out  such  things.  But  when  you 
are  older  I  hope  that  you  will  be  able  to  find  out 
such  things  yourselves. 


GEOMETRY.  9 

CHAPTEE    II. 

POINTSj    LINES,   AND    PLANES. 

1.  A  POINT  is  a  place  without  any  size.  When 
I  make  a  dot  to  mark  the  place  of  a  point,  that  is, 
to  show  where  the  point  is,  yon  must  not  think 
that  the  point  is  so  big  as  the  dot.  The  point  has 
no  size  at  all,  but  is  only  a  place  without  any 
size.  If  I  put  my  dot  in  the  right  place,  the 
point  will  be  exactly  in  the  middle  of  the  dot.  In 
common  talking,  we  sometimes  call  anything  that 
is  very  small  a  point ;  and  so  w^e  talk  of  the  point 
of  a  needle,  or  of  the  point  of  a  lead-pencil.  But 
in  Geometry  a  point  is  a  place  so  small  that  it  has 
no  size  at  all ;  neither  width,  nor  length,  nor  depth. 

2.  A  line  is  a  place  that  is  r-^ 
long  without  having  any  breadth  A  ^\^^.,^ 
or  thickness.     When  I  make  a  ^ 

long,  fine  stroke  with  a  pen  or  pencil,  or  with  a 
piece  of  chalk,  you  must  not  think  that  the  stroke 
of  pencil,  or  ink,  or  chalk-mark,  is  itself  the  line. 
I  only  make  it  to  show  where  the  line  is,  or  to  help 
you  imagine  a  line  ;  but  the  line  itself  is  the  mid- 
dle of  the  stroke  ;  you  cannot  see  it  any  more  than 

How  large  is  a  point  ?  How  can  you  mark  the  position  of  a 
point  ?  In  what  part  of  the  dot  is  the  point  supposed  to  be  ? 
What  is  the  difference  between  the  word  point  in  common  talk 
and  in  Geometry  ?  How  wide  is  a  line  ?  How  shall  we  mark 
a  line  ?    In  what  part  of  the  stroke  should  the  line  be  ?    What 


10  GEOMETRY. 

you  can  see  a  point,  for  it  is  only  a.  place  ;  and, 
although  it  has  length,  it  has  no  breadth  nor 
thickness. 

3.  The  ends  of  a  line  are  points.  You  may  fancy 
a  line  to  be  the  middle  of  a  very  fine  wire,  and  then 
you  will  easily  see  that  the  ends  of  it  are  points. 
j^  4.  When  the  point  of  my  pen  or  pencil 
I  moves  along  on  paper,  a  fine  stroke  of  ink 
or  of  pencil-mark  is  left  behind  it.  And 
that  may  help  you  fancy  a  real  point  mov- 
ing along  and  leaving  a  real  line  behind  it. 
It  will,  you  know,  be  only  fancy ;  because 
a  real  point  is  only  a  place,  and  a  place 
cannot  move.  But  it  is  a  good  way  to 
fancy  a  line  as  marked  out  by  the  track  of 
B  a  moving  point ;  that  is,  by  the  very  centre 
of  the  end  of  a  pencil.  It  will  help  you  very 
much  in  understanding  Geometry,  if  you  fancy  Cj 
line  as  the  track  of  a  moving  point. 

5.  A  plane  is  a  flat  surface,  like  the  floor,  or  the 
top  of  the  table,  or  like  your  slate.  I  need  not 
tell  you  any  more  exactly  what  a  surface  is,  and 
what  a  flat  surface  means  :  because  I  am  going  to 

is  the  end  of  a  line  ?  How  can  you  fancy  this  so  as  to  make  it 
like  a  needle-point  ?  If  a  point  could  move  and  leave  a  track 
behind  it,  what  would  that  track  be  ?  How  may  all  that  is 
described  in  the  first  thirty-five  chapters  of  the  book  be 
drawn  ? 


GEOMETRY.  11 

be  confined  to  one  plane  for  a  long  while.  I  mean 
that  for  a  good  many  chapters  I  shall  tell  you  only 
about  such  lines  as  can  be  drawn  upon  your  slate, 
or  upon  the  blackboard. 

6.  Now  you  have  studied  enough  for  one  lesson. 
If  you  understand  this  well,  you  have  made  a  very 
good  beginning  in  Geometry. 


CHAPTER    III. 

ABOUT   STRAiaUT   LINES   AND    CURVES. 

1.  A  LINE  that  is  not  bent  in  any  part  of  it  is 
called  a  straight  line.  If  we  fancy  a  point  moving 
in  a  straight  line,  we  shall  see  it  moving  always  in 
the  same  direction.  A  straight  line  is  the  shortest 
path  that  can  be  made  from  one  point  to  another. 
So,  when  vre  wish  to  tell  the  distance  from  one 
place  to  another,  we  measure  how  long  the  straight 
line  is  that  joins  the  two  places.  If  a  thread  is 
stretched  tight  across  a  table,  it  marks  a  straight 
line  across  the  table.  This  is  the  way  that  car- 
penters mark  a  straight  line,  by  rubbing  the  thread 
first  with  chalk;  and  gardeners  lay  out  garden- 
paths  and  beds  by  stretching  a  line. 

What  is  a  straight  line  ?  What  is  the  direction  in  which  a 
point  moves  when  moving  in  a  straight  line  ?  What  is  the 
shortest  path  from  one  place  to  another  ?     How  does  a  car- 


12  GEOMETRY. 

2.  A  curve  line  bends  in  every  part,  but  has  no 
sharp  corners  in  it.     And,  if  we  fancy  a  point 
moving  in  a  curve,  we  shall  see  it  all  the  time 
changing  its  direction,  but     ^ 
never  taking  sudden  turns. 
Perhaps  it  will  help  you 
understand  the  difference  between  a  straight 
line  and  a  curve,  if  I  draw  two  lines  at  the 
side  of  the  page.     I  think  you  will  under- 
stand, from  what  I  have  said,  which  is  the 
straight  line,  and  which  is  the  curve. 

3.  Now  I  want 
^"^^^-^^      _  _,    you  to  see  that    B 

D__^ ^^^"""^^^c;;^  two  straight  lines  can 

^"^^^  g  never  cut  across  each 
other  in  more  than 
one  place.  If  you  draw  only  two  lines  on  your 
slate,  and  each  line  is  straight,  they  cannot  cross 
each  other  in  two  places.  But  you  cannot  draw 
a  curved  line  that   you  cannot  ^^ — -\ 

cut,    at  least,    in   two   places.      /^^ — — - — b\ 
Try  to  draw,  on  your  slate,  Sb    ^  ^  I 

penter  mark  a  straight  line  ?  How  does  a  gardener  make  his 
paths  straight?  What  is  a  curve  line?  How  does  a  point 
move  in  a  curve  ?  In  how  many  places  can  one  straight  line 
cross  another  ?  Can  a  straight  line  always  cut  a  curve  in 
more  than  one  place  ?  In  how  many  places  can  a  straight  line 
always  cut  a  curve  ?    Let  the  teacher  draw  a  curve  upon  the 


GEOMETRY.  13 

curve  that  cannot  be  cut  in  two  places  by  one 
straight  line. 

4.  There  is  one  thing  more  that  I  want  you  to 
learn  at  this  lesson.  Whenever  a  straight  line 
joins  two  points  of  a  curve,  there  is  always  some 
point  on  the  curve   between  the  two   points   at 

which   the   curve  goes  in  the         ^^ -v,^ 

same  direction  as  the  straight      /^_^^ — """M^ 
line.     So  in  my  figure  you  see    ^^  I 

that  the  curve  at  c  goes  in  the  same  direction  as 
the  straight  line  A  B.  Draw  on  your  slate  figures 
of  curves,  and  cut  them  by  straight  lines,  and  you 
will  find  it  always  so.  This  seems  like  a  very 
simple  thing,  and  yet  it  is  a  very  useful  truth. 


CHAPTER    IV. 

ABOUT  ANGLES. 

1.  When  two  straight  lines  go  in  different  direc- 
tions, the  difference  of  their  directions  is  called  an 

blackboard,  cross  i^  by  a  straight  line,  and  then,  moving  the 
chalk  along  the  curve,  require  the  scholars  to  say  "now," 
whenever  the  chalk  is  moving  in  the  same  direction  as  the 
straight  line. 

What  is  an  angle  ?     On  what  does  the  size  of  an  angle  de- 
pend ?    Let  the  teacher  draw  angles  on  the  blackboard,  and 

2 


14 


GEOMETRY. 


B 


angle.  The  size  of  the  angle  depends,  then,  on 
the  difference  of  the  directions  of  the  lines,  and  not 
on  their  length.  So  that 
the  angle  which  I  havt^ 
marked  b,  is  larger  than 
that  which  I  have  marked 
A,  because  there  is  more  difference  in  the  direction 
of  the  two  lines  at  b,  than  of  the  two  lines  at  A. 

2.  The  point  where  two  straight  lines  meet,  or 
where  they  would  meet  if  we  fancied  them  drawn 
long  enough,  is  called  the  vertex  of  the  angle. 
The  vertex  of  the  a,ngles  a  and  b  is  not  marked 
down  ;  but  you  may  draw  two  straight  lines  meet- 
ing, and  the  point  where  they  meet  will  be  the 
vertex  of  the  angle  between  them. 

3.  When  two  straight  lines  cross  each  other, 
they  make  four  angles.     So,  in  the  figure  in  the 

margin,  we  have  two 
straight  lines  making 
the  four  angles,  A  E  c, 

A   E  D,    B   E   C,    BED. 

But  when  we  say  they 
the   angle   A  E  c,  we   have  to  fancy  the 


make 


ask  which  is  the  larger,  which  the  smaller,  etc.,  being  careful 
to  make  some  of  the  angles  without  vertices.  What  is  the  ver- 
tex of  an  angle  ?  Let  the  teacher  call  the  scholar  to  the  board 
to  point  out  the  vertices  of  the  angles  he  has  drawn.  When 
two  straight  lines  cross  each  other,  what  is  always  true  about 


GEOMETRY.  ^      15 

straight  line  D  c  going  in  exactly  the  opposite 
direction  to  that  in  which  it  goes  to  make  the 
angle  A  e  D. 

4.  If  we  fancy  that  the  line  A  b,  in  the  next  fig- 
ure, points  from  A  to  B,  while  the  line  c  D  points 
from,  c  to  Dj  the  two  lines  will  make  the  angle  E.  But 
if  the   line  A  B  points 

from  B  to  A,  they  wall  -^ 

make  the  angle  F.  When 

these    two    angles    are     ^ ^X^ 

equal,  each  is  called  a  ^ 

ri^^^ht  ano;le. 

5.  When  two  straight  lines  cross  each  other, 
the  opposite  angles  are  of  the  same  size.  I  mean 
that  in  this  figure  the 

angle  A  E   c  is  just      A> 
as  large  as  the  angle 
DEB,  and  the  angle 
A  E  D  just  as  large  as 
the  angle  c  E  b. 

6.  When  two  straight  lines  crossing  each  other, 
make  four  equal  angles,  each   angle  is  called  a 

the  angles  they  make  ?  What  is  a  right  angle  ?  What  ex- 
amples can  you  give  of  a  right  angle  ?  What  is  the  common 
name  for  the  vertex  of  a  right  angle  ?  The  teacher  must  be  very 
careful  not  to  let  the  child  confound  the  measure  of  an  angle 
with  either  the  length  of  the  sides,  or  area  of  the  opening 
between  them  ;  but  illustrate  and  explain  it  only  by  differ- 


16     .  GEOMETRY. 

right  angle.  Draw  two  lines 
on  your  slate  at  right  angles 
to  each  other.  The  side  of 
your  slate  is  at  right  angles  to 
the  bottom.  The  top  of  a  sheet 
of  letter-paper  makes  a  right 
angle  with  the  side.  The  sides  of  every  square 
corner  are  at  right  angles  to  each  other.  The 
vertex  of  a  right  angle  is  called  a  square  corner. 
When  two  square  corners  are  put  together,  the 
outside  edges  will  form  a  straight  line. 


CHAPTER    V. 

PARALLEL   LINES. 

1.  When  two  straight  lines  make  no  angle  with 
each  other,  or  when  they  make  an  angle  equal  to 
two  right  angles  with  each  other,  they  are  called 

parallel.    That  is 

y/^  to    say,    parallel 

/g  ^      lines  are  straight 

c ^^^ D    lines   that    point 

'y^  ■         in  the  same  di- 

rection, or  in  ex- 

ences  of  directions  ;  such  as  points  of  compass,  arrows,  turn- 
ing your  face  about,  etc.  When  two  square  corners  are  put 
too'ether,  how  do  the  outsides  run  ? 


GEOMETRY.  17 

actly  opposite  directions.     Try  whether  you  can 
draw  such  upon  your  slate. 

2.  When  two  straight  lines  are  parallel,  they 
are  just  as  far  apart  in  one  place  as  in  another. 
They  could  not  come  nearer  and  then  go  further 
apart  without  bending ;  but  a  straight  line  does 
not  bend  in  any  part.  And,  if  they  kept  coming 
nearer  until  they  met,  they  would  make  an  angle 
with  each  other,  and  the  point  where  they  met 
would  be  the  vertex  of  the  angle.  But  parallel 
lines  make  no  angle  with  each  other. 

3.  If  two  straight  lines  are  just  as  far  apart  in 
one  place  as  in  another,  they  are  parallel ;  they 
run  in  the  same  direction.  Try  whether  the  sides 
of  your  slate  are  parallel,  by  measuring  whether 
they  are  just  as  far  apart  at  the  top  of  the  slate  as 
at  the  bottom. 

4.  When  two  curves  are  everywhere  at  the  same 
distance  apart,  they  are  called  concentric  curves. 
Sometimes  they  are  called  par- 
allel curves  ;  but  this  is  not  so 
good  a  name  for  them  as  con- 
centric curves. 

Look  about  the  room,  or  out  of  the  window,  and  tell  me 
what  straight  lines  you  can  see.  Do  you  see  any  curve  lines  ? 
Any  lines  that  make  angles  with  each  other  ?  Can  you  show 
me  any  parallel  lines?  Any  concentric  curves?  What  are 
parallel  lines  ?  What  can  you  say  about  the  distance  apart  of 
2# 


. ^ 


18  GEOMETRY. 

5.  When  a  straight  line  crosses  two  parallel 
lineSj  it  makes  the  same  angles  with  the  one  as 

with  the  other. 
The  direction  of 
parallel   lines   is 

c -y^ D   alike,  and  so  the 

"^  '  .       diiFerence  of  their 

directions     from 
that  of  the  straight  line  must  be  alike. 

6.  If  a  straight  line  is  parallel  to  one  of  two 
parallel  lines,  it  is  parallel  to  the  other.  Draw 
now  two  parallel  lines  on  your  slate.  Draw  a 
third  line  parallel  to  one  of  your  first  pair,  and  it 
will  be  parallel  to  the  other.  All  three  of  the 
lines  will  point  in  the  same  direction. 

paraUel  lines  ?  When  a  straight  line  crosses  two  parallel  lines, 
what  can  you  say  about  the  angles  ?  (Let  the  teacher  beware 
of  forcing  a  child  to  repeat  the  reasoning  of  sections  two  and 
five.  To  the  teacher  the  reasoning  is  easier  than  the  concep- 
tions ;  to  the  child  it  is  just  the  reverse.)  How  can  you  tell 
whether  two  lines  are  parallel  ? 


GEOMETRY. 


19 


CHAPTER    VI. 

A   LITTLE   ABOUT   TRIANGLES. 

1.  A  TRIANGLE  is  a  figure 
bounded  by  three  straight 
lines.  They  are  very  sim- 
ple-looking things ;  and  yet  there  are  many 
curious  things  known  about  them  already.  And 
those  that  know  most  about  Geometry  tell  us  that 
no  one  has  yet  found  out  all  that  can  be  known 
about  them. 

2.  The  three  angles  of 
a  triangle  taken  togeth- 
er will  make  two  right 
angles.  You  can  try  it, 
if  you  like,  by  cutting  a 
triangle  out  of  paper,  with  a  pair  of  scissors.  Be 
very  careful  to  make  the  edges  straight.  Now 
cut  ofi*  two  of  the  corners  by  a  waving  line,  and 
lay  the  three  corners  of  the  triangle  carefully 
together.  The  outer  edges  will  make  one  straight 
line,  just  as  if  you  had  put 
two  square  corners  together. 
You  may  make  the  triangle  of 
any  shape  or  size,  and,  if  the 
edges   are   straight,  you   will 

Draw  triangles,  and  ask  ''  What  is  the  name  of  these  fig- 
ures?"    What  is  a  triangle?    Draw  a  right  triangle,  and, 


20  GEOMETRY. 

always  find  that  the  three  corners  put  together 
make  a  straight  line  with  their  outer  edges,  just  as 
two  square  corners  would  do.  And  this  is  what 
we  mean  by  saying  that  the  three  angles  of  a  tri- 
angle taken  together,  will  make  two  right  angles. 

3.  A  triangle  cannot  have  more  than  one  angle 
as  large  as  a  right  angle. 

4.  If  one  angle  in  a  triangle  is  a  right  angle, 
the  other  two,  put  together,  will,  of  course,  just  be 
equal  to  a  right  angle.  You  can  try  this  by  cut- 
ting paper  triangles  with  one  square  corner,  and 
then  cutting  off  the  other  corners  by  a  waving 
line,  and  putting  them  together. 

5.  If  one  side  of  a  triangle  is  longer  than 
another  side,  the  angle  opposite  the  longer  side  is 
larger  than  that  opposite  the  shorter  side.     Now 

look  at  the  fig- 
ure. The  side  a 
is  opposite  the 
angle  A,  and  the 
side  h  opposite  the  angle  b.  The  side  h  is  longer 
than  the  side  a  ;  and  from  this  we  may  know  that 
the  angle  b  is  larger  than  the  angle  A. 

6.  Now,  on  the  other  hand,  when  one  angle  in 

pointing  to  the  square  corner,  ask,  "What  angle  is  this  ? ' '  How 
i^iany  square  corners  can  a  triangle  ever  have  ?  Hovr  much 
do  the  three  angles  of  a  triangle  put  together  make  ?  If  one 
angle  is  a  right  angle,  how  much  do  the  other  two  put  to- 


GEOMETRY. 


21 


a  triangle  is  larger  than  another,  the  side  opposite 

the  larger  angle  is  longer  than  the  side  opposite 

the  smaller  angle. 

So,    if  we   know  ^ 

that  B   is   larger  ^ 

than  c,  we   may 

know  that  b  is  larger  than  c. 


CHAPTER    VII. 


MORE   ABOUT   TRIANGLES. 

1.  Suppose  that  we  found  two  sides  of  a  tri- 
angle to  be  just  equal  to  each  other,  what  should 
we  know  about  the  angles  ?  We  should  know  that 
the  angle  opposite 
one  side  was  just 
as  large  as  the 
angle  opposite  the 
other  side.  If  the  side  a  is  just  as  long  as  the 
side  c,  the  angle  A  is  just  as  large  as  the  angle  c. 


getlier  make.  If  one  side  of  a  triangle  is  longer  than  another, 
wliat  do  you  know  about  the  angles  ?  If  one  angle  is  larger 
than  another,  what  do  you  know  about  the  sides  ? 

When  we  know  that  two  sides  of  a  triangle  are  equal,  what 
do  we  know  of  the  angles  ?  When  we  know  that  all  the  sides 
of  a  triangle  are  of  the  same  size,  what  do  we  know  about  the 
angles?    When  we  know  that  two  angles  are  equal,  what  do 


22  GEOMETRY. 

2.  But  if,  on  the  other  hand,  we  know  that  the 
two  angles  are  equal,  we  shall  know  from  that 
that  the  two  sides  are  equal.  If  the  angle  A  is 
just  as  large  as  the  angle  c,  then  the  side  a  must 
be  just  as  long  as  the  side  c.  Such  a  triangle  is 
called  isosceles,  which  means  equal-legged. 

3.  Now  if  the  three  sides  of  a  triangle  are  each 
equal  to  each  other,  then  the  angles  are  equal  to 
each  other ;  and  if,  on  the  other  hand,  the  three 

angles  are  equal  to  each  other,  then  the 
sides  are  equal  to  each  other.  Such  a 
triangle  is  called  equiangular,  or  equi- 
lateral. Perhaps  this  drawing  will  help  you  to 
imagine  an  equilateral  triangle. 

4.  If  a  line  be  drawn  through  a  triangle  paral- 
lel to  one  side  of  the  triangle,  it  divides  the  other 
two  sides  in  the  same  proportion.  I  mean  that,  if 
in  such  a  triangle  as  A  b  c  we  draw  I)  E  parallel  to 

A  B,  then  c  D  will  be  the  same 
part  of  c  A  that  c  e  is  of  c  b. 
If  c  D  is  two  thirds  of  c  A,  then 
c  e  will  be  two  thirds  of  c  b. 
5.  Besides  what  I  have  al- 
ready told  you  about  the  last  figure,-^  that  is, 

we  know  about  the  sides?  When  we  know  that  the  three 
angles  are  equal  ?  What  is  an  equiangular  triangle  ?  What 
is  an  equilateral  triangle?  Every  equiangular  triangle  is 
also ?  And  every  equilateral  triangle  is  also ?  Let  the 


GEOMETRY.  23 

about  any  triangle  that  is  cut  in  two  by  a  line 
parallel  to  one  side, —  there  are  two  other  curious 
things  for  you  to  learn.  In  the  first  place,  A  b 
will  be  in  the  same  proportion  to  D  c  that  b  e  is  to 
E  c ;  and  in  the  second  place,  C  D  will  be  in  the 
same  proportion  to  c  E  as  a  c  is  to  B  c,  and  A  D 
will  also  be  in  the  same  proportion  to  e  b.  If  b  c 
is  three  quarters  of  A  c,  then  c  E  will  be  three 
quarters  of  c  D,  and  E  b  will  be  three  quarters 
of  A  D. 

6.  If  we  divide  one  side  of  a  triangle  into  equal 
parts,  and  then  draw  lines  through  the  points 
where  we  have  divided  the  side,  making  these 
lines  parallel  to  another  side  of  E 
the  triangle,  the  third  side  will  be 
divided  into  equal  parts.  Thus, 
if  the  side  A  b  is  divided  into 
equal  parts,  the  side  B  c  is  also 
thus  divided. 

7.  I  am  afraid  this  lesson  will  be  difficult  to 
understand ;  but  in  the  next  I  will  try  to  tell  you 
something  that  will  be  easier, 

teacher  now  copy  the  figure  of  section  tour  upon  the  blackboard, 
and  ask  'what  lines  are  in  the  same  proportion  as  c  d  and  c  a  ? 
What  in  the  same  as  c  d  to  c  e  ?  What  in  the  same  as  a  d  to 
DC?  Let  the  teacher,  also,  draw  parallel  lines  at  equal  dis- 
tances apart,  like  a  staff  of  music,  and  then,  stretching  a  string 
across  them,  show  the  scholars  that  the  lines  divide  the  string 
equally  in  whatever  direction  it  is  held. 


24 


GEOMETRY. 


CHAPTER    VIII. 

RIGHT   TRIANGLES. 

1.  If  you  do  not  remember  what  a  right  angle 
is,  you  must  turn  back  and  read  chapter  iv.,  sec- 
tion 6,  and  then  you  will  be  ready  to  go  on  with 
this  chapter. 

When  a  triangle  has 
a  right  angle  for  one  of 
its  angles,  it  is  called 
a  right  triangle.      The 

angle  a  b  c  is  a  right  angle,  and  the  triangle  ABC 

a  right  triangle. 

2.  Any  triangle  may  be  divided  into  two  right 
triangles,  by  drawing  a  line  through  the  vertex  of 
the  largest  angle  in  such  a  way  as  to  make  right 
angles  with  the  longest  side.     Thus,  if  b  is  the 

Q     largest  angle  in  the  tri- 


angle A 


b   c,    we    can 


draw  B  D  in  such  a  way 
as  to  make  the  angles  at 
D  right  angles ;  and  this  will  divide  the  triangle 
into  two  right  triangles,  abb  and  c  D  B. 

Now  draw  any  triangles  you  please,  upon  your 

What  is  a  right  angle  ?  What  is  the  cominon  name  for  the 
vertex  of  a  right  angle  ?  What  is  a  right  triangle  ?  How  can 
you  divide  any  triangle  into  two  right  triangles  ?    The  teacher 


GEOMETRY.  25 

slate,  and  trj  whether  you  cannot  always  divide 
them,  in  this  way,  into  two  right  triangles. 

3.  If  the  largest  angle  in  a  triangle  is  larger 
than  a  right  angle,  Ave  can  always  fancy  a  right 
triangle  added  to  it  in  such  a  vfay  as  to  make  the 
whole  figure  a  right  triangle.  If  B  be  larger  than 
a  right  angle,  we  can  ^ 
make  A  b  longer,  and  ^.,^^'^^"^^'' 
draw    c   D  down   in              ^^^^^--^"'^^       /    \ 

such   a    way   as    to^^--^""""^^ ^^ ip 

make  D  a  right  angle. 

Then  the  added  triangle  b  c  D  is  a  right  triangle, 
and  the  whole  figure  A  D  c  is  also  a  right  triangle. 
And,  by  taking  away  b  D  c  from  ADC,  we  shall 
have  ABC  left.  So  that  any  triangle  with  one 
angle  larger  than  a  right  angle,  like  ABC,  may 
be  fancied  as  the  difference  between  two  right  tri- 
angles, like  ADC  and  b  D  c. 

4.  If  we  divide  a  right  triangle  into  two  right 
triangles,  as  I  have  told  you  how  to  do  in  the  sec- 
ond section  of  this  chapter,  the  two  little  triangles 
will  be  of  exactly  the  same  shape  as  the  whole 
large  triangle. 

may  call  the  class  to  the  blackboard,  and  allow  them  to  draw 
triangles  and  divide  them  in  this  way.  To  what  kind  of  tri- 
angle can  you  add  a  right  triangle  so  as  to  make  the  whole 
figure  a  right  triangle  ?  Let  the  pupils  show  this  by  drawing 
them  on  the  blackboard.    "What  kind  of  triangle  can  be  fancied 


26  GEOMETRY. 

Thus,  if  B  is  a  right 
angle,  and  the  angles  at 
D  are  both  right  angles, 
then  the  three  triangles, 

A  B  c,  A  B  D  and  B  D  c,  are  all  of  exactly  the  same 

shape. 


CHAPTER    IX. 

SIMILARITY   AND    ISOPERIMETRY. 

1.  There  are  two  very  hard-looking  words  at 
the  head  of  this  chapter ;  but  they  are  not  hard  to 
understand.  Similarity  means  likeness  ;  and  in 
Geometry  it  means  the  having  the  same  shape. 
Isoperimetry  ^-^  means  the  being  of  the  same  size 
round  about. 

2.  When  two  bodies,  or  two  geometrical  figures, 
are  of  exactly  the  same  shape,  we  call  them  sim- 
ilar. When  two  figures  are  similar,  that  is,  of 
exactly  the  same  shape,  the  angles  of  one  figure 
are  exactly  equal  to  the  angles  of  the  other  figure, 


as  the  difference  between  two  right  triangles  ?  If  we  divide  a 
right  triangle  into  two  right  triangles,  what  do  you  know 
about  them? 

What  is  the  meaning  of  "  similar  "  figures  ?    What  is  true 
cf  the  angles  of  similar  figures  ?    Let  the  teacher  draw  two 

*  Isoperim'etry. 


GEOMETRY.  27 

and  all  the  sides  of  the  first  figure  are  in  the  same 
proportion  to  the  corresponding  sides  of  the  other 
figure.  I  told  you  that 
A  D  B  and  c  D  B  are  of 
.the  same  shape.  So 
that  the  angle  at  A  is 
just  as  large  as  the  angle  d  B  c,  the  angle  at  c 
just  as  large  as  D  B  A,  and  the  two  angles  at  D  are 
equal.  And  c  B  is  the  same  part  of  A  B  that  c  D 
is  of  D  B,  or  that  d  b  is  of  A  D.  All  this  is  meant 
by  saying  that  the  triangle  c  D  B  is  of  the  same 
shape  as  A  D  B. 

3.  If  we  find  that  the  three  angles  of  one  tri- 
angle are  just  equal  to  the  three  angles  of  another 
triangle,  we  may  know  that  the  three  sides  of  the 
first  triangle  are  in  the 
same  proportion  to  the 
corresponding  sides  of  the 
second  triangle,  and  that  the  triangles  are  similar. 
That  is  to  say,  that  if  the  three  angles  of  one  tri- 
angle are  equal  to  the  three  angles  of  another  tri- 
angle, the  sides  opposite  to  the  equal  angles  are  in 
the  same  proportion  to  each  other. 

4.  It  is  also  true  that  when  two  figures   are 

similar  quadrilaterals  on  the  board,  make  a  little  circle,  star, 
cross  and  accent,  in  the  four  angles  of  one,  and  bid  a  member 
of  the  class  come  and  put  corresponding  marks  in  the  angles 
of  the  other  figure,  thus  :  **  Who  will  mark  in  that  figure  the 


28 


GEOMETRY. 


similar,  any  two  sides  of  the  one  are  in  the  same 
proportion  to  each  other  that  the  corresponding 
sides  of  the  other  are  in  to  each  other.     Thus,  c  d 

is  the  same  part  of  c  b 

that  B  D  is  of  B  A.    And 

D  B  is  the  same  part  of 

D  A  that  D  c  is  of  D  B. 

5.  All  the  nice  calculations  of  engineers,  and 

machinists,  and  ship-builders,  and  navigators,  and 

astronomers,   are  made  by  help  of  similarity  of 

triangles. 

I  will  try  to  explain  to  you  one  single  instance 
in  Avhich  you  can  use  similarity  of  triangles.  Sup- 
pose that  a  house  stands  on  level  ground,  and  you 

wish  to  find  out  how  high 
it  is.  Put  a  stake  up- 
right in  the  ground,  any- 
where that  you  think 
best,  say  at  b.  Then  lay  your  head  close  to  the 
ground,  and  move  it  until  you  can  just  see  the  top 
of  the  house  over  the  top  of  the  stake.  Then 
measure  how  far  your  eye  is  from  the  bottom  of 
the  stake,  and  how  far  from  the  bottom  of  the 
house.     Then,  as  you  will  see  by  the  figure,  you 


angle  which  is  equal  to  this  that  I  marked  with  a  cross  ?  Who 
will  mark  the  one  equal  to  this  one  marked  with  a  circle?  " 
etc.,  etc.  Then  let  the  teacher  make  a  cross  through  two 
sides  of  one  quadrilateral,  and  say,  **  Who  will  mark  the  two 


GEOMETRY.  29 

will  have  two  triangles  of  the  same  shape,  and  the 
height  of  the  house  will  be  the  same  part  of  the 
distance  A  c,  that  the  height  of  the  stake  is  of  the 
distance  A  b.  If  the  height  of  the  stake  is  equal 
to  the  distance  of  your  eye  from  the  bottom  of  it, 
then  the  height  of  the  house  is  just  equal  to  the 
distance  of  your  eye  from  the  foundation  of  the 
house.  Another  way  of  finding  similar  triangles 
to  measure  a  house,  or  tree,  on  level  land,  is  by 
using  shadows.  The  shadow  of  the  stake  B,  Avhen 
the  sun  shone,  would  be  a  triangle.  The  stake 
would  be  one  side,  the  shadow  on  the  ground 
another,  and  the  third  side  would  be  the  edge  of 
the  shadow  in  the  air,  going  from  the  top  of  the 
stake  to  the  end  of  the  shadow  on  the  ground.  A 
similar  triangle  would,  at  the  same  time,  be  made 
by  the  shadow  of  the  tree  or  house.  So  that,  if 
you  should  measure  at  any  time  the  length  of  the 
shadow  of  the  stake,  and  the  length  of  the  shadow 
of  the  house,  you  could  tell  the  height  of  the 
house ;  because  the  height  of  the  house  would  be 
the  same  part  of  the  length  of  its  shadow  that  the 
stake  was  of  its  shadow. 

6.  When  it  is  just  as  far  round  one  figure  as  it 


sides  in  the  other  figure  that  are  in  the  same  proportion  as 
these  ?''  Next  let  him  draw  a  figure  of  a  liberty-pole,  and  a 
stake,  and  ask  the  child  to  explain  in  what  way  he  can  meas- 


30  GEOMETRY. 

is  round  another,  the  two  figures  are  called  iso- 
perimetrical,^  which  means  equal  round  about. 

If  one  triangle  measures,  on  its 
sides,  two  feet,  and  five  feet,  and 
six  feet,  its  perimeter,!  that  is,  the 
distance  round  it,  will  be  thirteen 
feet ;  because  two  and  five  and 
six  make  thirteen.  And  if  another  triangle  meas- 
ures on  its  sides,  three  feet,  and  six  feet,  and  four 
feet,  its  perimeter  will  also  be  thirteen  feet ;  be- 
cause three  and  six  and  four  make  thirteen.  So, 
these  two  triangles  will  be  isoperimetrical ;  but 
they  will  not  be  similar,  that  is,  they  will  be  of 
difierent  shapes. 

7.  Similar  figures  are  those  of  the  same  shape  ; 
isoperimetrical  figures  are  those  which  measure 
equally  round  about. 


CHAPTER    X. 

THE   SIZE   OF   TRIANGLES. 

1.  If  one  side  of  a  triangle  can  grow  longer  or 
shorter,  while  the  opposite  angle  opens  and  shuts, 

ure  tlie  height  of  the  liberty-pole  by  means  of  the  stake.    When 
are  two  figures  called  isoperimetrical  ? 

*  isoperimefricail.  t  Perim'eter. 


GEOMETRY. 


31 


as  though  its  vertex  were  a  hinge,  the  triangle  vail 
be  largest  when  that  angle  is  a  right  angle.  Thus, 
if  A  B  and  B  c  can- 
not be  changed  in 
length,  but  if  A  c 
is  like  an  India- 
rubber  cord,  and 
can  be  made  longer 
or  shorter  by  altering  the  angle  at  B,  then  the  tri- 
angle ABC  will  be  largest  when  the  angle  at  B  is 
a  right  angle. 

2.  You  can  show  this  very  prettily  in  this  way. 
Take  two  little  straight  sticks  and  hold  them  to- 
gether at  one  end  with  your  thumb  and  finger, 
while  you  spread  the 
other  two  ends  against 
the  edge  of  the  table. 
The  triangle  made  by 
the  two  sticks  and  the 
table-edge  will  be  largest  when  the  sticks  make  a 
right  angle  with  each  other. 

3.  If  one  side  of  a  triangle  cannot  change  in 
length,  and  if  the  other  two  can  only  change  ia 
such  a  way  as  to  keep  the  triangle  isoperimetrical 

If  two  sticks  lean  their  tops  togetlier,  how  must  they  b^ 
placed  to  make  the  space  between  them  and  the  ground  larg- 
est ?  It*  two  boards  be  nailed  together  to  make  a  pig-trough, 
what  angle  must  they  make  so  as  to  have  the  trough  hold 


AAA 


i;^^^ 

32  GEOMETRY. 

the  triangle  will  be  largest  when  these  two  sides 

are  of  equal  length.     Thus,  if  b  c  cannot  grow 

either  longer  or  shorter, 
and  if  A  B  grows  shorter 
exactly  as  fast  as  A  c 
grows  longer,  or  grows 
^    longer  exactly  as  fast  as 

A  c  grows  shorter,  the  triangle  will  be  largest  when 

A  B  is  equal  to  A  c. 

4.  You  may  show  this  in  a  very  pretty  way  by 
tying  the  ends  of  a  string  to  the  ends  of  a  straight 
stick  a  good  deal  shorter  than  the  string.  Then 
take  the  stick  in  one  hand,  and,  putting  one  finger 

of  the  other  inside  the  string, 

pull  it  tight.     The   triangle 

\  formed  between  the  string  and 

/  F^  ff\  /  ^\>^  the  stick  will  be  largest  when 
your  finger  is  in  the  middle 

of  the  string.     But  when  you  move  your  finger 

the  triangle  remains  isoperimetrical,  if  the  string 

does  not  stretch. 

5.  If  _we  suppose  that  all  three  sides  of  a  tri- 
angle can  change  their  length,  but  only  in  such  a 
way  as  to  keep  the  triangle  isoperimetrical,  then 
the  triangle  will  be  largest  when  the  three  sides 
are  equilateral.     That  is  to  say,  that  an  equilat- 

most  ?  If  a  tent  is  in<ade  simply  of  two  slant  sides  like  the 
roof  a  house,  how  must  we  pitch  it  so  as  to  have  most  room 
in  it? 


aEOMETRY.  33 

eral  triangle  is  the  largest  among  isoperimetrical 
triangles. 

6.  You  can  show  this  by  taking  a  long  piece  of 
string  and  tying  the  ends  together.  Put  into  this 
loop  one  finger  of  your  right  hand,  and  get  two 
playmates  each  to  do  the  same.  Stepping  back, 
you  can  pull  the  string  into  a  large  triangle,  and 
you  will  see  that  it  is  largest  when  your  fingers 
are  at  equal  distances,  and  the  triangle  equilateral. 
But  all  the  triangles  made  by  moving  the  fingers 
will  be  isoperimetrical,  because  the  string  remains 
of  one  length.  When  you  grow  •  older  you  can 
prove  these  things ;  but  now  you  can  best  show 
them  to  yourself  in  such  ways  as  this. 


CHAPTER    XI. 

DIFFERENT   KINDS   OF   TRIANGLES. 

1.  When  a  triangle  has  its  three  sides  equal, 
it  is  called  an  equilateral  triangle.  Equilateral 
means  equal-sided.  Every  equilateral  triangle  has 
its  angles  equal  to  each  other;  and  is,  therefore, 
called  equiangular.  But  equilateral  is  the  more 
common  name. 

Let  the  teacher  provide  sticks  and  strings  and  make  tri- 
angles, as  directed  in  sections  two,  four  and  six,  and  ask, 
*'  When  -will  this  triangle  be  largest  ?  " 


34  GEOMETRY. 

2.  When  a  triangle  lias  two  of  its'sides  of  the 
same  length  it  is  called  an  isosceles^  triangle. 
Isosceles  means  equal-legged.  The  angles  opposite 
the  equal  legs  are,  as  I  hope  you  remember,  equal 
to  each  other. 

An  equilateral  triangle  must  be  isosceles ;  but  an 
isosceles  triangle  may  not  be  equilateral.  Every 
horse  is  an  animal ;  but  not  every  animal  is  a 
horse. 

3.  Suppose  that  A  b  c  is  an  isosceles  triangle, 
and  that  a  is  equal  to  c.     Now,   if  we  draw  a 

line  B  h  from  B  to 
the  middle  of  A  c, 
it  will  divide  the 
triangle  into  two 
equal  right  triangles.  The  angle  at  b  is  divided 
exactly  in  the  middle,  the  angles  at  h  are  right 
angles,  and  we  could  fold  the  triangle  over 
on  the  line  B  i  as  a  hinge,  and  the  two  pieces 
would  fit  each  other  exactly. 

4.  If  a  triangle  has  no 
two  sides  equal,  it  is  called 
a  scalene  triangle.     Sca- 
lene means  lame,  or  limping. 

Set  the  class  at  the  blackboard,  and  ask  them  to  draw  an 
equilateral  triangle,  and  tell  what  they  know  about  its  angles 
and  sides.     An  isosceles  triangle.     A  scalene  triangle.     An 

*  isos'celes. 


GEOMETRY.  35 

5.  When  a  triangle  has  one  right  angle,  it  is. 
as  I  hope  you  remember,  called  a  right  triangle. 
We  can  have  isosceles  right  triangles,  and  scalene 
right  triangles ;  but,  of  course,  we  cannot  have 
equilateral  right  triangles. 

6.  When  neither  of  the  angles  of  a  triangle  is  ri 
right  angle,  the  triangle  is  called  an  oblique  *  tri- 
angle. Oblique  triangles  may  be  equilateral,  or 
isosceles,  or  scalene. 

isosceles  oblique  triangle.  Isosceles  right  triangle.  Scalene 
right  triangle.  Scalene  oblique  triangle.  Ask  them,  also, 
to  point  out  all  the  triangles  in  objects  within  sight,  and  name 
them  according  to  this  chapter. 


Keview  of  Triangles.  —  Let  the  scholars  now  turn  back 
to  chapter  vi.,  and  take  six  chapters  as  a  review-lesson,  and 
answer  the  following  questions,  and  others  selected  from  the 
questions  on  each  chapter. 

What  is  a  triangle  ?  If  the  three  corners  of  a  paper  tri- 
angle are  placed  together,  what  will  their  sum  be  ?  How  will 
the  outer  edges  run  ?  In  a  right  triangle  what  will  the  sum  of 
the  two  smaller  angles  be  ?  In  an  isosceles  right  triangle  what 
part  of  a  right  angle  will  each  smaller  angle  be  ?  How  are 
the  smallest  angle  in  a  triangle  and  the  shortest  side  in  the 
same  triangle  placed  ?  If  one  corner  of  a  triangle  be  cut  off 
by  a  line  parallel  to  the  opposite  side,  what  is  the  shape  of  the 
little  triangle  that  you  thus  make  ?  If  we  divide  a  right  tri- 
angle into  two  right  triangles  by  a  line  through  the  vertex  of 
tlie  right  angle,  what  will  be  the  shape  of  these  little  triangles  ? 
When  are  two  triangles  similar  ?  When  isoperimetrical  ?  When 

*  Obiek', 


36  GEOMETRY. 

CHAPTER    XII. 

QUADRANGLES. 

1.  When  a  figure  is  bounded  by  four  straight 

lines,  it  is  called  a  quadrangle.     Sometimes  it  is 

called  a  quadrilateral,  but  generally  a  quadrangle. 

-Q  2.  A  line  joining 

^^^^---'T""-^..^^^^  two  opposite    verti- 

^^^^rrlTL. _..../ _^(^  ces  is  called  a  diag- 

\.         /      ^^.^--'^"''^        onal.    In  this  figure, 
^si^--''^'^  the  dotted  line  A  c 

is   a    diagonal,    and 
the  dotted  line  b  d  is  another  diagonal. 

3.  Each  diagonal  divides  the  quadrangle  into 
two  triangles.  If  both  diagonals  are  drawn,  the 
quadrangle  is  cut  up  into  four  little  triangles ;  but 
those  are  not  the  triangles  of  which  I  am  speaking. 
The  dotted  line  A  c,  in  the  figure,  divides  the 
quadrangle  into  tw^o  triangles,  arc  and  c  D  A. 

only  one  side  of  a  triangle  can  alter  in  length,  how  shall  we 
make  the  triangle  largest  ?  When  two  sides  only  can  alter, 
but  the  triangle  is  kept  isoperimetrical,  how  is  it  made  larg- 
est ?    What  is  the  largest  of  all  isoperimetrical  triangles  ? 


What  is  a  quadrangle  ?  What  do  you  call  a  line  that  joins 
two  opposite  Tertices  ?  Into  what  two  figures  is  a  quadrangle 
divided  by  a  diagonal  ?  What  is  the  sum  of  the  angles  of  a 
triangle  ?  What  is  the  sum  of  the  angles  of  a  quadrangle  ? 
How  can  you  show  this  by  paper  figures?    Let,  now,  the 


GEOMETKY. 


37 


And  I  think  you  can  easily  see  that,  if  we  put  all 
the  corners  of  a  quadrangle  together,  they  will 


make  just  as  large  a  sum  as  if  we  put  all  the 
angles  of  two  triangles  together. 

4.  The  sum  of  the  four  an- 
gles of  any  quadrangle  is  equal 
to  four  right  angles.  But  if 
we  put  the  vertices  of  four 
right  angles  together,  the  sides 
of  the  angles  will  make  two  straight  lines  crossing 
each  other. 

And  if  we  make  a  paper  quadrangle,  cut  off  the 
corners  (by  a  waving  line,  so  as  not  to  become  con- 
fused), and  put  these  four  corners  together,  we 
shall  find  that  they 
fill  up  all  the  space 
around  the  corners. 
Compare  now"  the  let- 
ters in  these  four  an- 
gles with  the  letters  at  the  corner  of  the  first 
quadrangle  that  I  drew  for  you,  on  page  36. 

teacher  draw  quadrangles  and  diagonals  on  the  board,  and 
ask,  "  What  is  this  figure  ?     What  is  this  line? "  etc.    What 
4 


38 


GEOMETRY. 


In  cutting  paper  tri- 
angles and  quadrangles,  to 
show  that  the  corners  put 
together  make  two  or 
four  right  angles,  you 
must  be  careful  to  make 
the  sides  straight.  Cut 
off  the  corners  by  a  wav- 
ing line,  that  you  may  distinguish  the  edges  of  the 
cuts  from  the  edges  of  the  triangles  or  quad- 
rangles. 

5.  The  shape  of  a  triangle  cannot  be  altered, 
except  by  altering  the  length  of  at  least  one  side.. 
But  the  shape  of  a  quadrangle  can  be  altered 
without  altering  the  length  of  a  single  side,  by 
altering  the  angles.  The  diagonals  of  a  quad- 
rangle can  be  made  shorter  or  longer  without  short- 
ening or  lengthening  the  sides.  If  you  will  look 
J.  at  the  figure  you  will 

see  that  b  d  might 
be  pulled  out,  and 
A  c  crowded  togeth- 
er, so  as  to  alter  the 
shape  of  the  quad- 
rangle very  much,  without  altering  the  length  of 
its  sides  at  all. 


is  the  only  figure  that  is  strong  and  stiff?    In  what  figures 
can  the  angles  be  altered  without  altering  the  sides  ?    What  is 


GEOMETRY. 


39 


6.  Take  two  -willow  twigs,  or 
two  thin  rolls  of  paper,  such  as 
are  used  for  lamp-lighters,  and 
bend  one  into  a  triangle,  and  one 

into  a  quadrangle.  You  will 
find  that  the  triangle  is  stiff, 
and  that  the  quadrangle  is 
not. 

7.  If  you  have  ever  noticed  the  frame  of  a 
house,  you  have  seen  that  the  carpenter  puts 
braces  in  the  corners.  With- 
out braces  the  timbers  would 
only  make  quadrangles,  and 
so  would  have  no  stiffness 
except  the  stiffness  of  the 
joints.  But,  by  putting 
braces  he  makes  triangles, 
which  cannot  be  pressed  out 
of  shape  without  being  broken. 


the  use  of  a  brace  ?    Let  the  teacher  proyide  twigs  or  lamp- 
'lighters  for  the  illustration  of  section  six. 


40  GEOMETRY. 

CHAPTER    XIII. 

PAR  ALLELO  GRAMS. 

1.  Whex  a  quadrangle  has  its  opposite  sides 
parallel  it  is  called  a  parallelogram.  So  that 
there  are  in  every  parallelogram  two  sets  of  par- 
allel sides. 

2.  The  opposite  angles  of  a  parallelogram  are 
equal.     If  this  figure  is  a  parallelogram,  so  that 


A  D  is  parallel  to  b  c,  and  D  c  parallel  to  A  b,  then 
the  angle  at  d  is  equal  to  the  angle  at  B,  and  the 
angle  at  A  is  equal  to  that  at  c. 

3.  A  diagonal  divides  a  parallelogram  into  two 
equal  triangles.  If  you  could  turn  the  triangle 
ABC  roundj  so  as  to  put  the  point  b  exactly  on 
the  point  i),  and  the  line  B  c  upon  the  line  D  A^ 
then  the  line  B  A  would  lie  upon  the  line  d  c,  and 
one  triangle  would  exactly  cover  the  other.  You 
can  try  this  by  cutting  a  parallelogram  of  paper^ 

What  is  a  parallelogram  ?  How  many  sets  of  parallel  sides 
in  a  parallelogram  ?  How  many  sets  of  equal  angles  ?  Which 
angles  of  a  parallelogram  are  equal  ?  Let  the  class  draw  par- 
allelograms on  the  blackboard,  and  show  which  angles  are 


GEOMETRY.  41 

and  cutting  it  in  two  diagonally.  But  you  must 
be  very  careful  to  have  the  opposite  sides  exactly 
parallel. 

4.  And  this  shows  you  that  in  every  parallel- 
ogram the  opposite  sides  are  equal.  In  the  figure. 
A  D  is  equal  to  B  c,  and  A  b  to  D  c.  If  the 
opposite  sides  of  a  quadrangle  are  parallel,  they 
are  equal. 

5.  And  if  a  quadrangle  has  its  opposite  sides 
equal,  they  are  also  parallel.  If  I  should  find  a 
quadrangle  such  as  this  figure,  and,  by  measuring, 


find  that  A  D  and  B  c  are  just  equal,  and  also  that 
A  B  and  I)  c  are  equal  to  each  other,  I  should  know 
that  these  equal  lines  are  parallel. 

6.  When  a  quadrangle  is  squeezed  flatter  with- 
o'ut  altering  the  leno:th  of  its  sides,  it  alters  the 
size,  as  well  as  the  shape,  of  the  quadrangle.  The 
quadrangle  will  be  largest  when  the  opposite  an- 
gles, added  together,  are  equal  to  two  right  angles. 

equal?  Draw  a  diagonal  in  your  parallelogram.  How  does 
it  divide  tlie  figure  ?  How  can  you  tell  whether  a  figure  is  a 
parallelogram  ?  How  must  I  lay  four  sticks  of  unequal  length 
on  the  fioor  so  as  to  enclose  the  largest  space  ?  How  must  I 
4# 


42  GEOMETRY. 

In  this  figure,  the 
angles  B  and  D  added 
Pj  together  would  be 
more  than  two  right 
angles.  We  could, 
then,  make  the  quad- 
rangle larger,  without  altering  the  length  of  its 
sides,  by  drawing  b  and  d  apart,  and  crowding  A 
and  c  together,  until  the  two  angles,  B  and  d, 
added  together,  made  two  right  angles.  The 
angles  A  and  c  would,  also,  added  together,  then 
be  equal  to  two  right  angles,  and  the  quadrangle 
would  be  as  large  as  we  could  make  it  without 
changing  the  length  of  the  sides. 

7.  When  a  parallelogram 
is  put  into  its  largest  form, 
as  the  opposite  angles  are 
always  equal,  and  the  two 
are  equal  to  two  right  angles,  each  of  the  four 
angles  will  be  a  right  angle. 

place  them  when  the  sticks  can  be  divided  into  two  couples  of 
equal  length  ?  Let  the  teacher  show  the  child  that  there  are 
two  different  answers  to  the  last  question  ;  one  a  parallelo- 
gram, the  other  not.  And,  giving  a  child  four  unequal  sticks, 
let  him  try  how  many  quadrangles,  all  of  the  largest  possible 
sizCj  he  can  make. 


GEOMETRY.  43 

CHAPTER    XIY. 

RECTANGLES   AND    SQUARES. 

1.  When  each  angle  of  a  parallelogram  is  a 
right  angle,  the  figure  is  called  a  rectangle.  This 
is  a  very  common  figure  in  all  sorts  of  things  that 
men  make.  The  panes  of  glass  in  the  windows, 
and  the  windows  themselves,  the  doors  and  the 
panels  in  them,  the  sides  of  the  room,  the  leaves 
of  books,  sheets  of  paper,  and  many  other  things, 
are  usually  made  in  the  shape  of  rectangles.  But 
in  the  things  that  were  not  made  by  men  there 
are  very  few  rectangles ;  they  are  scarcely  to 
be  found  even  in  crystals ;  coarse  salt  and  iron 
pyrites  ^'  being  the  only  common  things  in  which 
the  Creator  has  used  the  rectangle. 

2.  When  all  the  sides  of  a  rec- 
tangle are  equal,  the  figure  is  called 
a  square.  And,  as  the  corners  of  a 
square  are  all  right  angles,  so  a 
right  angle  is  sometimes  called  a 
square  corner. 

What  is  the  geometrical  name  of  the  figure  of  a  pane  of 
glass  ?  Name  all  the  objects  you  can  think  of  which  are  rec- 
tangular in  shape.  What  natural  objects  present  the  form  of 
a  rectangle  ?    If  you  found  a  rectangular  piece  of  stone  or 

*  Pyri'tes. 


44  GEOMETRY. 

iiiiinin|ni'iimri|       3.  When  carpenters  wish  to 


mark  a  right  angle  on  their 
boards  or  timber,  they  use  a 
simple  tool,  which  I  dare  say 

you  have  seen,  and  which  is  called  a  carpenter's 

square. 

4.  But,  in  Geometry,  the  word  square  is  only 
used  to  mean  a  rectangle  with  equal  sides. 

5.  I  suppose  you  know  the  way  in  which  men 
measure  how  long  a  thing  is,  w^ith  a  rule,  or  yard- 
stick, or  tape.  They  measure  how  many  inches,  or 
feet,  or  yards,  or  miles,  it  will  take  to  stretch 
along  by  the  side  of  the  thing  they  are  measuring. 

6.  Men  measure  surfaces,  such  as  painting,  or 
carpeting,  or  fields,  by  finding  out  how  many 
squares,  whose  sides  are  each  one  inch,  or  one  foot, 
or  one  yard,  it  will  take  to  cover  the  surface  which 
they  want  to  measure.  When  a  carpet-dealer  sells 
so  many  yards  of  oil-cloth  or  oil-carpeting,  he 
means  that  the  carpet  could  be  cut  in  such  a  way 
as  to  make  just  that  number  of  squares,  one  yard 
on  a^  side.  When  a  man  says  that  there  are  so 
many  feet  of  land  in  his  door-yard,  he  means  that 
it  would  take  just  that  number  of  square  pieces  of 
paper,  one  foot  on  a  side,  to  cover  his  yard. 

iron,  would  you  easily  believe  that  men  had  not  cut  it  into  that 
shape  ?  What  is  a  carpenter's  square  ?  What  is  a  square  cor- 
ner ?    "What  is  the  geometrical  meaning  of  the  word  square  ? 


GEOMETRY.  45 

7.  It  is  very  easy  for  any  one  that  knows  the 
multiplication  table,  to  measure  a  rectangle.  For 
the  number  of  square  inches, 
or  yards,  in  a  rectangle  is 
found  by  multiplying  the 
number  of  inches,  or  yards, 
in  the  breadth  by  the  num- 
ber in  the  width,  and  this  will  give  the  numoer  of 
squares  in  the  rectangle. 

If  you  have  learned  to  count,  you  will  see  by 
this  figure  that  a  rectangle  four  feet  wide  and 
seven  feet  long,  could  be  divided  into  twenty- eight 
squares,  each  being  a  foot  on  a  side.  We  can 
divide  it  into  seven  rows,  of  four  square  feet  in  a 
row,  or  into  four  rows,  seven  square  feet  in  a  row. 

8.  When  we  wish  to  measure  a  parallelogram 
that  is  not  a  rectangle,  we  have  only  to  multiply 
the  length  by  the  breadth ;  because  every  parallel- 
ogram is  exactly  the  same  size  as  a  rectangle,  of 
the  same  length  and  breadth,  would  be. 

You  see  in  the  figure  that  the  rectangle  A  B  c  D 
is  of  the  same  length  and  breadth  as  the  parallel- 
How  do  men  measure  the  length  of  things  ?  "What  do  they 
mean  by  saying  a  hundred  feet  of  land  ?  How  do  they  find 
out  the  number  of  square  feet  or  square  inches  in  a  rectangle  ? 
In  a  parallelogram?  Let  the  teacher  copy  the  last  figure 
on  the  blackboard,  and  ask  the  children  how,  if  the  rectangle 
and  parallelogram  were  made  of  paper,  they  could  cut  the 


46 


GEOMETEY. 


ogram  A  b  F  E. 
And  they  are 
of  exactly  the 
same  size.  If 
you  cut  the 
triangle  c  A  E 
off  one  end  of  the  rectangle  and  place  it  at  the 
other  end,  it  will  just  cover  the  triangle  b  D  F. 

9.  When  I  say  that  we  multiply  the  length  of  a 
parallelogram  by  its  breadth  in  order  to  find  its 
measure,  I  mean,  of  course,  that  we  must  multiply 
the  nwnher  of  inches  in  the  length  by  the  num- 
her  of  inches  in  the  breadth  in  order  to  find  the 
number  of  square  inches,  that  is,  squares  an  inch 
on  a  side,  that  it  would  take  to  cover  the  parallel- 
ogram. Numbers  are  the  only  things  that  can  be 
multiplied. 


I 


CHAPTER    XV. 

TRIANGLES   AND    RECTANGLES. 

1.     Every  triangle  may  be  imagined  as  half 
of  a  parallelogram.     If  we  had  a  triangle  A  b  f, 

rectangle  to  make  it  cover  the  parallelogram,  and  how  the 
parallelogram  to  make  it  cover  the  rectangle. 
How  do  men  measure  triangles?    How  do  they  measure 


aEOMETRY. 


47 


we  could  draw 
A  E  parallel  to 
B  F,  and  F  E 
parallel  to  B  A, 
and  that  would 
make  a  paral- 
lelogram just  twice  as  large  as  the  triangle. 

2.  And  thus  we  may  find  the  measure  of  a  tri- 
angle ;  that  is,  we  may  find  how  many  squares  of 
one  inch  on  a  side,  or  of  one  foot  on  a  side,  it 
would  take  to  cover  the  triangle  exactly.  We 
need  only  multiply  any  one  side  of  the  triangle 
by  the  distance  to  a  parallel  line  drawn  through 
the  opposite  vertex.  This  will  give  us  the  measure 
of  a  parallelogram,  and  half  of  that  will  give  us 
the  size  of  the  triangle. 

S.  In  this  way  men  measure  surfaces  of  every 
shape,  by  dividing  them  into  triangles,  and  then 
finding  out  how  large  each  tri- 
angle is.  You  may  draw  upon 
your  slates  figures  of  any  num- 
ber of  straight  sides,  and  then 
divide  them  into,  triangles  by 
drawing  diagonals.  You  can 
divide  the  same  figure  into  different  sets  of  tri- 
angles by  drawing  difierent  diagonals. 


other  surfaces  than  parallelograms  ?    How  long  ago  did  Py- 


48  GEOMETRY. 

4.  In  every  right  triangle  the  square  made  on 

the  side  opposite  the  right 
angle  is  just  as  large  as  the 
squares  on  the  other  two 
sides,  put  together.  If,  in 
my  figure,  the  triangle  A  b  c 
has  a  right  angle  at  c,  the 
square  on  the  side  A  b  will 
be  just  as  large  as  the  square 
on  A  c  added  to  the  square 
on  c  B. 

5.  The  side  of  a  right  triangle  opposite  the 
right  angle  is  called  the  hypotenuse,  and  the  other 
sides  are  called  the  legs.  So  that  what  I  have 
already  told  you  may  be  repeated  in  different 
words.  The  square  on  the  hypotenuse  is  equiv- 
alent to  the  sum  of  the  squares  on  the  legs. 
This  is  called  the  Pythagorean  proposition,  be- 
cause it  was  found  out  by  a  geometer,  who  lived 
more  than  two  thousand  years  ago,  whose  name 
was  Pythagoras. 

6.  This  Pythagorean  proposition  gives  us  a 
good  way  of  trying  whether  an  angle  is  a  right 
angle.  If  the  sum  of  the  squares  on  two  sides  of 
a  triangle  is  just  equal  to  the  square  on  the  third 
side,  we  may  know  that  the  angle  opposite  this 
third  side  is  a  right  angle. 

thagoras  live  ?    What  did  he  discoTer  about  right  triangles  ? 


GEOMETRY.  49 

Now,  a  square  is  a  rectangle  as  broad  as  it  is 
long ;  so  that,  to  find  how  large  the  square  on  a 
leg  would  be,  we  need  only  multiply  the  number 
of  inches  in  the  length  of  the  leg  by  itself.  If 
you  should  measure  the  sides  of  a  triangle,  and 
find  that  one  side  was  three  inches,  and  another 
four  inches,  and  the  third  five  inches  ;  then,  if  you 
know  the  multiplication  table,  you  would  know 
that  the  squares  on  the  sides  would  be  nine 
inches,  and  sixteen  inches,  and  twenty-five  inches. 
Moreover,  if  you  add  nine  to  sixteen,  it  makes 
twenty-five.  So  that,  in  a  triangle  whose  sides 
are  three,  four  and  five  inches,  the  side*  of  five 
inches  is  a  hypotenuse,  and  the  opposite  angle  a 
right  angle. 

7.  If  you  understand  the  last  section,  you  will 
also  understand  how  the  Pythagorean  proposition, 
that  the  square  on  the  hypotenuse  is  equivalent 
to  the  sum  of  the  squares  on  the  legs^  is  very 
useful  to  carpenters  and  other  persons  who  wish 
to  make  square  work.  Suppose  you  were  making 
a  frame  of  a  house,  and  wanted  the  timbers  A  b 
and  A  c  to  make  a  right  angle  at  A.  You  might 
measure  three  feet  from  A  to   B,  and   four   feet 


How  can  you  tell  whether  a  triangle  is  a  right  triangle  ?  How 
do  carpenters  make  the  corner  of  a  frame  square  ?    How  do 

5 


60 


GEOMETRY. 


from  A  to  c,  and  then 
alter  the  angle  at  A,  by 
moving  one  of  the  tim- 
bers until  a  stick  of  five 
feet  long  would  just  reach 
from  c  to  B.  Instead  of 
three,  four  and  five,  for  large  frames,  they  take 
six,  eight  and  ten  feet. 

8.  Another  use  that  carpenters  make  of  the 
Pythagorean  proposition  is,  in  cutting  braces  for 
frames.  They  measure  the  same  number  of  feet 
from  A  to  B  and  from  A  to  c ; 
and  then  for  the  brace,  B  c, 
they  measure  just  as  many 
times  seventeen  inches  as 
there  are  feet  in  A  b.  This 
makes  the  brace  a  very  little 
too  long ;  because  a  square 
on  seventeen  inches  is  a 
very  little  more  than  twice  as  large  as  a  square 
foot.  But  carpenters  like  the  brace  to  be  a 
very  little  too  long,  because  hammering  the  tim- 
bers together  makes  the  brace  indent  the  tim- 
ibers  a  little ;  and,  if  the  brace  were  not  a  very 
"Kttle  too  lonnr  at  first,  it  would  be  a  little  too  short 


they  cut  braces  for  square  corners  ?    How  large  is  a  square 
built  on  the  diagonal  of  a  square  ?    How  large  is  a  square 


GEOMETRY.  51 

after  it  had  indented  the  timbers ;  and  that  would 
make  A  less  than  a  right  angle. 

9.  The  square  on  the  diagonal  of  a 
square  is  just  twice  as  large  as  that 
square.  This  you  will  see  in  the  fig- 
ure marked  a. 

A  square  with  its  corners  in  the 
middle    of    the    sides    of    another 
square  is  just  half  as  large  as  that 
square.     This  you  will  see  in  the 
figure  marked  B. 

If  we  fancy  a  square  cut  into  four 
right  triangles  by  two  diagonals  (as  at 
c),  we  can  fancy  each  little  triangle 
turned  over  on  its  own  little  hypote- 
nuse, which  will  make  the  figure  at  b  a  square  just 
twice  as  large  as  that  at  c. 

You  may  take  a  square  piece  of  paper,  as  at  B, 
fold  each  corner  over  to  the  centre  of  the  square, 
as  at  c,  then  unfold  it  again,  as  at  b,  and  I  think 
this  will  make  you  understand  this  section. 

whose  corners  are  in  the  middle  of  the  sides  of  another 
square  ?  Let  the  children  draw  right  triangles  and  squares 
upon  the  sides,  and  tell  the  relative  size  of  the  squares  accord- 
ing to  the  Pythagorean  proposition.  The  teacher  should 
require  this  proposition,  as  written  in  italics,  to  be  committed 
to  memory,  as  it  is  the  most  useful  proposition  in  the  fifteen 
chapters. 


52  GEOMETRY. 

CHAPTER    XVI. 

CIRCLES. 

1.  We  have  had  fifteen  chapters  about  straight 
lineSj  and  about  figures  bounded  by  straight  lines. 
I  think,  therefore,  that  you  will  be  glad  to  learn  a 
little  about  curves. 

2.  A  curve  is,  as  I  have  told  you,  a  line  that 
bends  in  every  part  of  it,  but  has  no  sharp  corners. 

^^  If  a  boy  were  wheeling  a  barrow 
in  a  large,  level  field,  just  after 
a  light  snow,  the  middle  of  the 
track  of  his  wheel  would  be  a  line  that  you  could 
easily  follow  with  your  eye. 

If  the  boy  went  all  the  time  in  the  same  direc- 
tion, he  would  make  a  straight  line ;  but  if  he 
kept  turning  a  little  all  the  while  to  the  right  or 
left,  he  would  make  a  curve. 

3.  If  the  boy  kept  on  all  the  time  turning  in 
the  same  direction,  just  as  fast  as  he  began  to 
turn,  the  track  of  the  wheel  would  come  round 

What  is  a  curved  line  ?  What  is  a  straight  line  ?  How 
could  you  mark  a  straight  line  with  the  wheel  of  a  wheelbar- 
row ?  How  would  you  mark  out  a  curve  with  the  wheel  ?  If 
a  line  bends  equally  in  every  part,  how  long  can  it  be  ?  (Only 
long  enough  to  return  into  its  own  beginning.)  What  is  a 
curve  called  that  has  no  ends,  and  bends  equally  in  every 
part  ?     But,  if  it  bends  equally  in  every  part,  and  yet  has  two 


aEOMETRY.  58 

into  itself,  and  make  what  is  called 
the  circumference  of  a  circle.  A 
curve  that  bends  equally  in  every 
part  is  called  a  circumference ;  and 
a  figure  bounded  by  a  circumfer- 
ence is  called  a  circle. 

4.  Parts  of  a  circumference  are 
called  arcs.  The  word  arc  used  to 
mean  a  bow;  and  bows  are  shaped 
something  like  an  arc. 

5.  If  there  was  a  tree  in  the  middle  of  a  field, 
and  the  boy  should  keep  all  the  time  at  the  same 
distance  from  the  tree,  he  would  come  round  to  the 
place  he  started  from,  and  the  track  of  his  wheel 
would  bend  equally  in  every  part.  He  would,  in 
fact,  go  round  in  the  circumference  of  a  circle. 

6.  You  will  see,  in  the  figure  of  a  circle  in  sec- 
tion three,  a  dot  in  the  middle.  It  shows  the 
place  where  we-  suppose  the  tree  to  stand.  And 
in  every  circle  there  is  a  point  in  the  middle, 
equally  distant  from  every  part  of  the  circum- 
ference. This  point  is  called  the  centre  of  the 
circle.     It  is  not  always  marked  by  a  dot,  but  the 

ends,  what  is  it  called  ?  When  I  swing  the  door  on  its  hinges, 
in  what  curve  does  the  end  of  the  latch  move  ?  Where  is  the 
centre  of  the  arc?  When  a  stone  hangs  by  a  thread,  and 
swings  backward  and  forward  over  a  straight  line  on  the  floor, 

5^ 


54  GEOMETRY. 

place  is  always  there  whether  it  is  marked  or 
not. 

7.  A  straight  line  drawn  from 
the  centre  of  a  circle  to  its  circum- 
ference is  called  a  radius.  All  the 
radii  of  the  same  circle  are  equal. 
In  this  figure  there  are  three  radii 
marked  by  black  lines. 

8.  We  can  fancy  a  circle  made  by  a  radius 
swinging  all  round  a  centre.  And  you  may  draw 
very  nice  circles  by  holding,  with  one  hand,  a 
thread  fast  to  one  spot  on  your  slate,  or  black- 
board, while,  with 
your  other  hand, 
you  hold  the  other 
end  of  the  string 
and  your  pencil 
together,  and  draw 
a  circumference, 
keeping  the  string  stretched.  You  can  also  draw 
them  with  one  hand,  by  resting  your  little  finger 
on  the  centre  of  the  circle,  and  being  careful  to 

what  curve  does  the  stone  move  in  ?  Where  is  the  centre  of 
the  arc  ?  What  line  does  the  thread  mark  out  ?  Let  the  teacher 
actually  open  and  shut  the  door,  and  swing  a  plumb,  while 
asking  these  questions.  Keep  the  child  constantly  in  the  habit 
of  seeing  the  perfect  geometrical  forms  suggested  by  the  ma- 
terial appearances.  Let  them  draw  circles,  and  their  radii,  on 
the  board. 


GEOMETRY.  55 

keep  the  end  of  your  crayon  always  at  the  same 
distance  from  it.  This  way  only  answers  for  the 
blackboard.     . 


CHAPTER    XVII. 

MORE    ABOUT    CIRCLES. 

1.  Rectangles  and  circles  are  the  most  com- 
mon figures  in  all  manufactured  things.  They 
are  the  easiest  figures  to  make  exact,  and  the  most 
convenient  when  made.  You  have  already  noticed 
how  common  rectangles  are  in  what  men  make. 
Circles  are  almost  as  common.  Buttons  and  door- 
knobs, plates,  pans  and  wheels,  are  circular.  But 
in  natural  things. —  in  things  made  by  the  great. 
Creator, —  circles  are  much  more  common  than 
rectangles.  The  sun,  the  moon  and  stars,  the 
buds  of  many  flowers,  the  eyes  of  animals,  and 
some  little  animals  themselves,  are  nearly  in  the 
shape  of  a  circle. 

2.  To  draw  circles,  men  have 
what  is  called  a  pair  of  compasses, 
that  open  and  shut  like  a  pair  of 
tongs.  They  put  one  point  down 
on  the  paper  hard  enough  to  hold  it 


What  is  the  figure  of  a  cent  ?     What  other  things  can  you 
t«li  me  of  in  the  shape  of  a  circle  ?    Is  the  circle  or  the  rec- 


56  GEOMETRY. 

still  in  the  centre  of  the  circle,  and  then  move  the 
other  leg  round  lightly,  just  bearing  on  hard 
enough  to  mark  the  circumference.  The  circle 
will  be  larger  or  smaller,  as  the  compasses  are 
opened  wider,  or  are  more  nearly  shut. 

You  can  make  a  pair  of  compasses,  to  draw  cir- 
cles on  the  ground,  by  putting  one  small  nail 
through  two  bits  of  shingle,  and 
whittling  the  ends  to  a  point.  The 
old  Greeks,  of  Pythagoras'  time, 
and  afterwards,  used  to  study  and 
teach  Geometry  with  the  help  of 
figures'  drawn  on  smoothly-spread 
sand. 

3.  To  make  things  circular-shape,  men  some- 
times draw  a  circle  first,  with  compasses,  and  then 
cut  the  thing  to  that  shape.  But  generally  they 
use  a  different  method.  They  use  a  turning-lathe, 
or  something  that  works  on  the  same  plan.  In  ^ 
turning-lathe  a  block  of  wood  or  piece  of  iron  is 
made  to  turn  steadily  round  on  a  steel  point  that 
marks  the  centre,  while  the  chisel,  that  cuts  the 
wood  or  iron,  is  held  steadily  at  the  same  distance 

tangle  oftener  found  in  natural  objects  ?  What  instrument  do 
men  have  to  draw  circles  with  ?  How  can  you  make  a  sort  of 
eom passes  ?  What  did  the  old  Gi-eeks  use  for  a  blackboard  and 
chalk  ?  Did  you  ever  see  a  turning-lathe  ?  A  potter's  wheel  ? 
A  tinman  at  work  cutting  out  round  pieces  ?    How  did  he  cut 


GEOMETKY.  57 

from  the  steel  point,  so  that  the  wood  or  iron  that 
is  too  far  from  the  centre  is  cut  off  as  it  goes  past 
the  chisel.  On  a  potter's  wheel  a  lump  of  clay  is 
made  to  turn  round,  while  its  centre  remains  in 
one  place ;  and  thus  the  potter  makes  the  crockery 
round.  The  bottom  of  tinned  pans  is  cut  out  by 
a  sort  of  sciss(3rs  that  is  fastened  at  the  right  dis- 
tance from  a  point  around  which  the  piece  of 
tinned  plate  is  made  to  turn. 

Sometimes  boys  make 
a  circle  of  leather,  for  a 
plaything.  Put  an  awl 
through  a  bit  of  leather 
into  a  board ;  then  stick 
a  sharp  knife  through  the  leather  into  the  board, 
slanting,  with  the  cutting-edge  down  ;  the  blade  at 
right  angles  to  a  line  joining  it  to  the  awl.  Now 
pull  the  leather  round  the  awl,  under  the  knife, 
and  you  will  cut  out  a  circle.  Put  a  strong  string 
through  the  awl-hole  in  the  leather,  with  a  large 
knot  at  the  end  of  it,  and  you  will  have  a  curious 
toy.  Wet  the  leather  and  press  it  under  your  foot 
upon  a  flat  stone,  and  you  can  lift  the  stone  by  the 
string.      But  be  careful  not  to  swing  the  stone 

them  ?  Did  you  ever  see  a  blacksmith  bend  a  wagon-tire  ? 
Why  does  passing  the  tire  through  the  three  rollers  make  a 
circle  of  it  ?  How  does  a  cooper  make  a  hoop  round  ?  Let  the 
teacher  draw  a  circle  with  radius,  cord,  and  diameter,  and 


58 


GEOMETRY. 


about ;  for  nobody  has  a  right  to  put  other  people 
in  danger. 

All  these  ways  of  making  a  circle  are  somewhat 
alike.  But  there  is  another  way, —  to  bend  wire,  or 
things  of  the  kind,  equally  at  every  part.  Thus, 
the  blacksmith  passes  an  iron  bar  between  three 
rollers,  that  bend  each  part  of  the  tife  equally,  and 
thus  bring  it  into  a  circular  form  for  a  wagon-tire. 
4.  A  straight  line  joining  the 
ends  of  an  arc,  is  called  a  chord. 
Any  straight  line  going  across 
a  ckcle,  having  both  ends  in  the 
circumference,  is  a  chord.  All 
the  lines  marked  A  c  and  b  c.  m 
this  figure,  are  chords. 

5.  The  longest  chord  that  we 
can  have  is  the  one  that  goes 
through  the  centre  of  the  circle. 
It  is  called  the  diameter,  and  is 
just  twice  the  length  of  a  ra- 
dius. 

6.  If  we  divide  a  circumference  into  six  equal 
arcs,  the  chord  of  each  arc  is  just  as  long  as  a 
radius.      So   that,  if  you  draw  a  circle   on  the 


ask  -what  the  name  of  each  line  is.  What  is  the  longest  chord 
we  can  have  ?  If  we  divide  the  whole  circumference  into  six 
equal  arcs,  how  long  is  the  chord  of  each  arc  ?  How  can  you 
show  it  with  a  pair  of 'Compasses  ? 


GEOMETRY. 


59 


ground  with  your  shingle  com- 
passes, you  will  find,  if  you  are 
careful  neither  to  open  nor  close 
your  compasses,  that  they  will 
step  round  the  circumference  in 
exactly  six  steps. 


CHAPTER    XVIII. 

MEASURING   ANGLES. 

1.  If  the  vertex  of  a  right  angle  is  at  the  cen- 
tre of  a  circle,  the  lines  forming  the  angle  cut  off 
one  quarter  of  the  circumference,  whether  the 
circle  is  large  or  small.  So 
with  any  angle  whose  vertex 
is  at  the  centre  of  a  circle ; 
its  sides  always  cut  off  just 
that  part  of  a  circumference 
which  the  angle  is  of  four 
right  angles.  If  it  takes  eight 
of  these  little  angles  to  make  one  right  angle,  it 
will  take  eight  of  the  little  arcs  to  make  a  quarter 
of  a  circumference,  whether  on  a  large  or  small 
circle. 

How  many  degrees  of  arc  make  a  quarter  of  a  large  cir- 
cumference ?  How  many  degrees  make  a  quarter  of  a  small 
circumference  ?    If  you  put  the  vertex  of  a  right  angle  at  the 


60  GEOMETRY. 

If  we  want  to  tell  how  large  an  angle  is,  we 
tell  what  part  of  a  ckcumference  it  would  cut  off 
if  its  vertex  were  at  the  centre. 

2.  And,  in  order  that  we  may  easily  tell  the 
parts  of  a  circumference,  we  fancy  every  circum- 
ference as  divided  into  quarters,  and  then  each  of 
these  quarters  divided  into  ninety  equal  parts^ 
which  ai'e  called  degrees  of  arc.  So  that  we  tell 
how  large  an  angle  is  by  telling  how  many  of 
these  degrees  would  be  cut  off  between  its  sides  if 
its  vertex  were  at  the  centre  of  a  circle. 

3.  If  the  vertex  of  a  right  angle  is  placed  at 
the  centre  of  a  circle,  its  sides  cut  off  a  whole 
quai'ter  circumference,  and  the  right  angle  is, 
therefore,    sometimes   called  an   angle  of  ninety 

degrees.  Half  a 
right  angle,  like 
the  angles  of  an 
isosceles  right  tri- 
angle, is  called  an  angle  of  foii:y-five  degrees. 

4.  As  each  right  angle  is  an  angle  of  ninety 
de^Jirees,  two  ri^ht  angles  too-ether  will  make  an 
angle  of  twice  ninety,  that  is,  of  one  hundred  and 
eighty  degi'ees. 

centre  of  a  circle,  how  many  degrees  of  arc  will  its  sides  cut 
off  ?  If  an  angle  is  an  angle  of  forty-five  degrees,  what  part 
of  a  right  angle  is  it  ?  Have  you  ever  studied  Arithmetic  ? 
Do  you  know  how  much  nine  times  ten  make  ?    Three  times 


GEOMETRY.  61 

5.  You  may  take  any  point  in  a  straight  j^ 
line,  as  the  point  c  in  the  line  a  b,  and 
£yicy  the  line  as  making  an  angle  of  one 
hundred  and  eighty  degrees  with  itself  at 
fliat  point.  An  angle  is  the  diflFerence  of 
direction  of  two  lines.  Now  you  cannot 
tell  whether  I  moved  my  pen  from  A  to  b, 
or  from  B  to  A ;  and  so,  if  you  like,  you 
may  fimcy  that  1  moved  it  from  A  to  c,  and 
then  from  b  to  c ;  that  is,  you  may  &ncy 
the  line  meeting  itself  at  c ;  that  is,  going  in 
opposite  directions  on  ^h  side  of  C;  diat  is, 
making  an  angle  of  two  right  angles  at  c  ;  that  is, 
the  line  makes  an  angle  of  one  hundred  and  eighty 
degrees  with  itself  at  c. 

6.  You  remember,  I  hope,  that  the  three  angles 
of  a  triangle  taken  together  are  equivalent  to  two 
right  angles.  We  can  now  say  the  same  thing  in 
other  words ;  we  may  say  that  the  sum  of  the 
three  angles  of  a  trian^e  is  one  hundred  and 
eighty  degrees. 

7.  In  an  equilateral   triangle,  you   remember 

thirty  ?  How  many  angles  of  10^  must  be  put  together  to 
make  a  ri^t  angle  ?  How  many  degrees  m  one  third  of  a 
ri^t  angie?  What  angle  does  a  straight  line  make  with 
itsdf  ?  How  many  degrees  do  the  three  angles  of  a  triangle 
added  together  make  ?  How  many  d^rees  in  each  angle  of 
&n  equilateral  triangle  ? 

6 


62  GEOMETRY. 


that  the  three  angles  are  equal  to  each 
other.  Each  of  them  is  then  an  angle 
of  sixty  degrees,   because   three  times 


sixty  is  one  hundred  and  eighty. 


CHAPTER    XIX. 

CHORDS. 

1.  If,  instead  of  putting  the  vertex  of  an  angle 
at  the  centre  of  a  circle^  we  put  it  in  the  circumfer- 
ence, it  will  take  in  an  arc  just  twice  as  large  as 
it  would  with  its  vertex  in  the  centre. 

2.  If  one  angle  has  its  ver- 
tex at  the  centre  of  a  circle^ 
as  at  c  in  the  figure,  anu 
another  has  its  vertex  in  the 
circumference,  as  at  A,  and  if 
B  ^        the  sides  of  both  these  angles 

go  through  the  circumference  at  the  same  places, 
D  and  B,  then  one  angle  is  just  half  as  large  as  the 
other.  The  angle  at  A  is  only  half  as  large  as 
that  at  c. 

An  angle,  with  its  vertex  at  the  centre,  is  measured  by  the 
arc  between  its  sides.  How  is  the  angle  measured  when  its 
vertex  is  in  the  circumference?  How  is  an  angle  between  two 
chords  measured,  when  the  vertex  is  in  the  circumference? 


GEOMETRY. 


63 


3.  We  can  say  the  same  thing  in  other  words. 
Two  chords  starting  from  the  same  point.  A,  in 
the  circumference  of  a  circle,  make  an  angle  of 
just  half  as  many  degrees  as  there  are  in  the  arc 
n  B  between  their  other  ends.  The  arc  d  b  is  not 
drawn  at  all ;  but  you  can  easily  fancy  it  there. 

4.  Now  the  arc  d  b  would  be  of  the  same 
length,  wherever  the  point  A  were  placed,  and 
that  will  make  you  understand  the  next  figure. 

5.  All  the  angles  that  can  be 
drawn  in  one  arc,  that  is,  with 
their  vertices  in  the  arc,  and 
their  sides  going  throu^  the 
ends  of  the  arc,  are  of  the  same 
size.  The  angles  at  c,  c,  c,  c, 
are  all  four  equal  to  each  other. 
Each  one  is  measured  by  half  the  arc  that  is  not 
drawn  between  A  and  B. 

6.  This  gives  you 
a  curious  way  of 
drawing  an  arc  of  a 
circle.  Drive  two 
pins  in  a  board,  as 
at  A  and  B,  and  then 


Can  any  of  the  class  draw  a  figure,  and  explain  how  a  curve 
in  a  railroad  is  laid  out  ?  Can  you  show  how  an  arc  may  be 
drawn  by  a  card  and  two  pins  ?  When  the  card  has  a  square 
corner,  how  large  is  the  arc  ?    Can  any  one  with  a  ruler  and 


64 


GEOMETRY. 


move  a  bit  of  card,  with  straight  edges,  in  such  a 
way  as  to  keep  the  corner  thrust,  as  far  as  it  will 
go,  between  the  pins.  The  corner  c  will  move  in 
the  arc  of  a  circle. 

7.  If  equal  angles  have  their  vertices  at  the 

same  point  in  the  cir- 
cumference, they  will 
cut  off  equal  arcs. 
That  is,  if  the  angles 
at  A  are  all  equal  to 
each  other,  the  arcs 
at  B,  B,  B,  are  equal 

^  to  each  other. 
This  is  the  way  in  which  railroads  are  laid  out 
in  curves.  The  engineer  measures  equal  angles 
from  one  point,  as  A,  and  equal  chords,  as  at  B,  B,  B, 
and  then  the  rails  are  laid  to  go  round  as  arcs  to 
those  chords. 

8.  There  are  several  other  ways  of  drawing  cir- 
cles without  using  compasses.  I  will  tell  you  one 
more,  which  you  will  find  very  useful  if  you  ever 
want  to  lay  out  curved  paths  in  a  garden,  or  do 
anything  of  that  kind.     Take  a  straight   stick. 


a  piece  of  chalk  show  a  way  of  staking  out  a  circular  garden- 
path  ?  (Use  the  ruler  for  the  stick  of  section  eighth,  and 
chalk  dots  for  stakes.)     Can  you  tell  me  how  to  try  whether 


aEOMETEY. 


65 


A  E  B,  and  drive  into  the  ground 

two  stakes,  one  at  A.  and  the  other 

at  D,  making  A  D  about  half  the 

length  of  the  stick.     Keeping  one  -^^ 

end  of  the   stick  at  A,  move  the 

other  end  round  until  the  middle,  E, 

is  as  far  from  D  as  you  think  best  to  put  it.    Then 

drive  a  stake  at  B.     Now  put  one  end  of  the  stick 

at  D,  and  let  the  middle,  e,  be  just  as  far  from  the 

stake  B  as  it  was  before  from  D.      Drive  a  new 

stake  at  the  other  end  of  the  stick.     Thus  you  can 

go  on,  driving  stakes  as  far  as  you  wish  to  go. 

The  size  of  the  circle  will  be  made  greater  by 

making  the  distance  D  E  smaller. 

9.  If  the  arc  a  c  B  is  just 
half  a  circle,  then  the  other 
arc  A  B  is  a  half-circle,  and 
the  angle  A  c  B  is  measured  by 
half  a  half-circle,  and  is  a  right 
angle.  If  the  corner  of  the 
card  in  section  six  of  this  chap- 
ter is  a  square  corner,  then  the  arc  will  be  a  half- 
circumference. 

10.  If,  in  the  last  figure,  A  c  is  a  right  angle, 


aD  angle  is  a  right  angle  without  using  the  Pythagorean  prop- 
osition ?  Invite  the  scholar  to  draw  a  figure  and  explain.  If 
he  cannot,  draw  it  for  him,  and  show,  with  shingle  compasses, 


66 


GEOMETRY. 


A  B  will  be  a  diameter,  and  the  middle  of  it,  D, 
will  be  the  centre  of  the  circle,  and  will  be  just  as 
far  from  c  as  it  is  from  A  or  B. 

This  gives  us  a  very  pretty  way  of  trying 
w^hether  an  angle  is  a  right  angle  or  not.  If  I 
want  to  know  whether  A  c  b  is  a  right  angle,  I 
will  draw  any  line  A  b  across  it ;  and,  if  half  A  B 
will  just  reach  from  the  middle  of  A  b  to  c,  we 
may  know  that  c  is  a  right  angle.  If  it  does  not 
reach,  c  is  less  than  a  right  angle  ;  if  it  more  than 
reaches,  c  is  more  than  a  right  angle. 


CHAPTER    XX. 


CHOKDS   AND    TANGENTS. 


1.  When  two  chords  do  not  touch  each  other, 
the  angle  between  them  is  measured  by  half  the 
difference  of  the  arcs  between 
them.  The  angle  made  by  the 
chords  A  B  and  c  D  is  measured 
J^  by  half  the  difference  between 
the  arcs  A  D  and  B  c.  That  is 
to  say,  the  angle  between  a  b 
and  c  D  is  half  as  large  as  the 


if  you  have  no  other,  how  to  divide  the  hypotenuse  and  apply 
the  test  of  section  ten. 

When  tvro  chords  do  not  touch  each  other,  how  is  the  angle 


GEOMETRY. 


67 


difference   of  the   two   angles   at   e,  A  E  D   and 

0  E  B. 

2.  We  may  imagine  the  chords  lengthened  into 
straight  lines  going  outside  the  circle  until  they 
meet ;  and  it  will 
not  alter  the  size 
of  the  angle.  So 
that  the  angle  A  c  b 
is  measured  by  half 
the  difference  of  the  arcs  between  its  sides,  wher- 
ever we  place  the  circle. 

3.  If,  in  the  last  figure,  we  move  the  circle 
back  until  the  circumference  touches  the  vertex  c, 
then  the  smaller  arc  has  become  of  no  size  at  all, 
and  the  difference  between  it  and  the  large  arc  is 
equal  to  the  whole  of  the  large  arc.  Then  the 
angle  is  measured  by  half  the  large  arc,  which  is 
just  the  same  thing  that  you  learned  in  the  last 
chapter. 

4.  If  we  take 
the  circle  farthest 
from  the  vertex 
in  the  last  figure, 
and  lift  it  up  till 
it  just  touches  the 


measured  ?  Can  you  draw  a  figure  on  the  blackboard,  and 
explain  this  more  fully  ?  When  two  straight  lines  go  through 
a  circle  and  meet  outside  the  circle,  how  is  the  angle  between 


68      .  GEOMETRY. 

line  A  c,  at  the  point  D,  as  in  this  figure,  then  the 
larger  arc  meets  the  smaller  arc  just  at  D. 

5.  The  line  A  c  is  now  called  a  tangent.  Tan- 
gent means  a  toucher ;  and  the  line  A  c  just  touches 
the  circumference  without  cutting  it. 

6.  If  now  we  move  the  line  b  c  without  chang- 
ing its  direction,  until  it  stands  in  the  place  of 
E  D,  the  small  arc  has  become  nothing ;  so  that 
the  angle  A  D  B  is  measured  by  half  the  arc 
between  its  sides, 

7.  In  other  words,  the  angle 

s\         between  a  chord  and  a  tangent 

"•n        at  one  end  of  the  chord  is  meas- 

y         ured  by  half  the  arc  between 

them.    That  is  to  say,  it  is  half 

as  large  as  the  angle  made  by  two  radii  to  the  ends 

of  the  chord. 

8.  When  the  chord  is  a  diameter,  the  angle  is 
measured  by  half  of  half  the  circle ;  that  is,  the 
angle  is  a  right  angle.  A  tangent  at  the  end  of  a 
diameter  must  always  be  at  right  angles  to  the 


them  measured  ?  Can  you  dra^v  a  figure,  and  explain  that  ? 
What  is  meant  by  a  tangent  to  a  circle  ?  How  is  the  angle 
between  a  chord  and  a  tangent  measured  ?  Draw  on  the  board 
a  circle  with  a  chord,  and  a  tangent  at  the  end  of  it.  Draw 
radii  to  the  ends  of  -the  chord.  Show  me  which  two  angles  in 
that  figure  should  be  the  one  just  double  the  other.  What  is 
the  angle  made  by  a  diameter  with  a  tangent  at  the  end  of  it  ? 


aEOMETRY.  69 

diameter.  And  you  will  easily  see  that  it  must 
be  so  with  a  radius.  A  tangent  at  the  end  of  a 
radius  is  at  right  angles  to  the  radius. 

9.  There  can  be  tangents  to  other  curves  as  well 
as  to  circles.  A  straight  line  that  just  touches 
a  curve ;  or,  if  the  curve  winds,  a  straight  line 
going  through  a  point  in  a  curve  in  the  same 
direction  as  the  curve  at  that  point,  is  called  a 
tangent  to  the  curve. 


CHAPTER    XXI. 

MORE  ABOUT  CHORDS  AND  TANGENTS. 

1.  When  two  chords 
cross  each  other,  the  an- 
gle they  make  is  meas- 
ured by  half  the  sum  of 
the  arcs  between  their 
ends.  That  is,  if  two 
straight  lines  cross  each 
other  inside  of  a  circle,  their  angle  is  measured  by 


By  a  radius  with  a  tangent  at  the  end  of  it  ?     What  is  a  tan- 
gent to  any  curve  ? 

How  do  you  measure  the  angle  of  two  chords  that  cross  each 
other  ?    How  is  it  when  the  chords  are  diameters  ?  If  a  chord 


70  GEOMETRY. 

half  the  siun  of  the  arcs  between  them ;  if  they 
cross  outside  the  circle,  the  angle  is  measured  by 
half  the  difference  of  the  arcs. 

2.  When  the  chords  are  both  diameters,  the  arcs 
are  equal,  and  half  the  sum  is  just  one  arc :  so 

that  the  angle  is  measured 
by  one  arc  ;  and  that,  you 
know,  is  the  first  thing 
that  you  learned  about  the 
measure  of  an  angle, — that 

an  angle,  with  its  vertex  in  the  centre,  is  measured 

by  the  arc  between  its  sides. 

3.  Arcs  are  parts  of  circumferences.  Parts  of 
other  curves  are  called  arcs  of  those  curves  ;  and 
straight  lines  joining  the  ends  of  those  arcs  are 
called  chords  of  those  arcs.  When  we  speak  of  an 
arc,  we  mean  a  piece  of  a  circle ;  and  if  we  wish 
to  speak  of  a  piece  of  an  ellipse,  we  call  it  an  arc 
of  an  ellipse.  You  will  learn  what  an  ellipse  is, 
after  a  while. 

4.  In  every  arc  of  any  curve  there  must  be 
at  least  one  place  at  w^hich  the  tangent  is  parallel 
to  the  chord  of  that  arc.     Now  turn  back  to  chap- 


of  an  arc  or  any  kind  of  curve  be  moved,  keeping  it  pax^allel 
to  its  first  position,  until  it  cuts  off  no  arc,  what  does  the  chord 
become  ?  If  I  go  to  a  place  exactly  north  from  me,  and  yet  at 
no  part  cf  my  journey  travel  north,  what  must  have  been  true 


GEOMETRY.  71 

ter  III.,  section  fourth, 
and  you  will  find  that 
what  I  have  now  told 
you  is  just  the  same  as 
that,  only  in  different  language. 

5.  If  the  chord  of  any  arc  of  any  curve  be 
moved,  keeping  it  parallel  to  its  first  position,  it 
will  cut  off  either  a  longer  or  else  a  shorter  arc. 
If  moved  one  way,  it  cuts  off  a  longer  arc ;  if 
moved  the  other  way.  it  cuts  off  a  shorter  arc.  If 
moved  so  as  to  cut  off  a  shorter  arc,  we  can  move 
it  so  far,  keeping  it  still  parallel  to  its  first  posi- 
tion, that  it  will  cut  off  no  arc  at  all,  but  be 
tangent  to  the  curve  at  that  point  where  the  curve 
goes  in  the  same  direction  as  the  chord.  You  can 
easily  perform  this  process  of  moving  a  chord,  and 
keeping  it  parallel  to  itself,  by  drawing  any  curve 
you  choose  on  your  slate,  and  moving  a  stretched 
thread  across  it. 

6.  Let  us  now  go  back  to  circles.  If  a  radius 
be  drawn  through  the  middle  of  a  chord,  it  will  be 


of  my  road?  (It  must  have  turned  a  comer  in  it.)  If  I  travel 
over  a  winding  road  without  corners,  and  find  myself  at  night 
in  a  place  just  east  of  my  starting,  what  do  you  know  of  the 
direction  of  my  road  ?  (It  must  have  gone  east  at  some  point 
of  the  way.)  What  can  you  tell  me  about  the  radius  that 
passes  through  the  middle  of  a  chord  ?    What  of  a  straight 


72 


GEOMETRY. 


at  right  angles  to  the  chord^  and  it 
will  end  at  the  middle  of  the  ai^c. 
Let  us  suppose  that,  in  this  figure, 
"^the  radius  c  D  goes  through  the 
middle  of  the  chord  A  b.  Then,  I 
say  that  the  angles  at  E  will  be 

right  angles,  and  the  arc  A  d  will  be  equal  to  the 

arc  D  B. 

7.  On  the  other  hand,  if  we  draw  a  straight 
line,  through  the  middle  of  a  chord,  at  right 
angles  to  the  chord,  it  will  pass  through  the  centre 
of  the  circle. 

8.  This  gives  you  an  easy  way  to  find  the  centre 
of  a  circle  when  you  have  an  arc  of  the  circle. 

^,<Z^     "^       You  have   only  to   draw  two 

^/^.^^-.-^Pt'^*"^^^^^    chords    not    parallel    to   each 

y  other    (the   larger    the    angle 

h  they  make  the  better),  and  then 

draw  lines  through  the  middle 

of  each  chord  at  right  angles  to  the  chord.     As 

both  these  lines  pass  through  the  centre  of  the 

circle,  the  centre  must  be  at  the  point  where  the 

lines  cross  each  other. 

9.  If  you  can  get  a  card,  such  as  business  men 


line  at  right  angles  to  a  chord,  through  the  middle  of  it?  If 
you  find  an  arc  drawn  on  the  blackboard,  how  can  you  find 
the  centre  of  it  ? 


GEOMETRY.  73 

have  advertisements  printed  on.  you  can  use  the 
longer  side  as  a  ruler  by  which  to  make  a  straight 
line  ;  and,  by  setting  the  short  side  carefully  on 
this  line,  you  can  draw,  by  the  long  side,  a  line  at 
right  angles  to  your  first  one.  You  can  probably 
find  an  old  printed  card  by  asking  for  it,  and  it 
will  serve  as  a  ruler  and  square.  Then  you  can 
draw  short  arcs  by  your  eye,  and  find  the  centres 
by  section  eight. 


CHAPTER    XXII. 

INSCRIBED    POLYGONS. 

1.  When  a  triangle  has 
its  vertices  in  a  circumfer- 
ence, it  is  said  to  be  in- 
scribed in  the  circle.  In 
other  words,  w^hen  the  sides 
of  a  triangle  are  chords  in  a 
circle,  the  triangle  is  said  to  be  inscribed  in  the 
circle.     A  b  c  is  an  inscribed  polygon. 

2.  If  we  want  to  put  a  circle  round  a  triangle, 
so  that  the  triangle  shall  be  inscribed  in  the  circle, 
we  can  easily  do  it  by  remembering  that  the  sides 

What  is  meant  by  an  inscribed  triangle  ?  Where  do  you  say 
that  the  vertices  of  an  inscribed  triangle  are  ?  What  are  the- 
sides  ?    If  you  find  a  triangle  ready  drawn,  how  can  you  find 

7 


74  GEOMETRY. 

of  the  triangle  will  be  chords  in  the  circle,  and  so 
we  can  find  the  centre  of  the  circle,  to  put  one  leg 
of  our  compasses  at,  by  section  eight  of  the  last 
chapter.     We  have  only  to  draw  lines  at  right 

angles  to  the  middle  of 
\wo  sides  of  the  trian- 
gle, and  the  point  where 
these  lines  cross,  at  c, 
will  be  the  centre  of  a 
circle,  whose  circumfer- 
ence will  pass  through  the  three  vertices  of  the 
triangle.  Try  this  with  triangles  drawn  on  the 
ground,  and  you  will  then  find  that,  by  putting 
one  foot  of  your  shingle  compass  at  c,  you  can 
draw  a  circumference  through  the  vertices. 

3.  If,  on  the  other  hand,  we  wish  to  inscribe  a 
circle  in  the  triangle,  so  that  each  side  of  the  tri- 
angle shall  be  tangent 
to  the  circle,  we  must 
draw  lines  dividing 
two  of  the  angles  of 
the  triangle  into  halves,  and  the  point  where  these 
lines  cross  each  other  will  be  the  centre  of  the 
circle. 

the  centre  of  a  circle  whose  circumference  will  pass  through 
the  Tertices  ?  How  will  you  find  the  centre  of  a  circle  to 
which  the  sides  of  the  triangle  will  be  tangent  ?  What  is 
jQi^ant  by  a  circle  inscribed  in  a  triangle?    How  can  yon 


I 


GEOMETRY.  75 


4.  But  you  do  not  know,  perhaps,  how  to  divide 
an  angle  into  halves.  Suppose,  then,  that  cab 
is  the  angle  you  wish  to 
divide.  Measure  A  b 
and  A  c  of  equal  lengths. 
Put  one  foot  of  a  pair 
of  compasses  at  b,  and  with  the  other  foot  scratch 
a  little  arc  near  what  you  think  is  the  middle  of 
the  angle.  Now  put  one  foot  at  c,  and,  with  your 
compasses  open  exactly  as  wide  as  before,  make 
another  arc  crossing  the  first.  A  straight  line 
from  A  through  the  points  where  the  arcs  cross 
will  divide  the  angle  into  two  equal  parts. 

5.  The  largest  triangle  that  can  be  inscribed  in 
a  circle  is  an  equilateral  triangle.  So  that  there 
are  two  kinds  of  triangles  in  which  the  equilateral 
triangle  is  largest ;  namely,  the  isoperimetrical, 
and  those  inscribed  in  one  circle. 

6.  You  remember,  I  hope,  that  the  radius  of  a 
circle  is  the  chord  of  a  sixth  part  of  the  circum- 
ference. If  you  draw  an  arc  long 
enough  to  draw  a  chord  of  the 
same  length  as  a  radius,  and  then 
draw  radii  to  the  ends  of  this 
chord,  you  will  make  an  equilateral,  and,  there- 
divide  an  angle  into  halves  ?  Let  the  pupils  illustrate  as  they 
recite,  by  drawings  on  the  blackboard.  What  is  the  largest 
triangle  that  can  be  put  in  a  given  circle  ;  that  is,  in  a  circle 


76  GEOMETRY. 

fore,  equiangular  triangle.  Each  angle  will  be 
one  third  of  two  right  angles ;  because  the  three 
are  equal  to  each  other,  and  the  three  together  are 
equal  to  two  right  angles.  If  three  of  them  would 
be  equal  to  two  right  angles,  six  would  be  equal  to 
four  right  angles.  And  so  the  corners  of  six  such 
triangles  would  fill  up  all  the  space  about  the 
centre  of  a  circle,  and  the  six  chords  would  just 
go  round  the  circle. 

7.  If  you  wish  to  draw  the  largest  triangle  that 
you  can  in  a  circle,  you  must  open  your  compasses 
just  as  wide  as  they  would  be  to  draw  the  circle, 
and  then  step  six  times  round  the  circumference, 
marking  the  points  where  the  feet  of  the  compasses 
step.  Then  join  every  other  one  of  these  marks 
by  three  straight  lines,  and  they  will  make  an 
inscribed  equilateral  triangle. 

8.  Do  you  understand  what  I  mean  when  I  say 
that  the  radius  is  equal  to  the  chord  of  sixty 
degrees  ? 


that  you  find  already  drawn  ?  How  long  is  the  chord  of  sixty 
degrees?  What  part  of  a  circumference  is  sixty  degrees? 
How  do  you  draw  an  equilateral  triangle  in  a  circle  ? 


GEOMETRY.  77 

CHAPTER    XXIII. 

MORE   ABOUT   INSCRIBED    POLYGONS. 

1.  Any  figure  bounded  by  straight  lines  is  a 
polygon ;  and,  if  its  vertices  are  in  a  circumfer- 
ence, it  is  an  inscribed  polygon, 

2.  A  polygon  of  three  sides  is  called  a  triangle ; 
of  four  sides  a  quadrangle  ;  of  five  sides  a  penta- 
gon ;  of  six  sides  a  hexagon. 

3.  Any  polygon  of  more  than  three  sides  can 
have  its  angles  altered  without  altering  the  length 
of  its  sides.  You  remember,  I  hope,  how  we  illus- 
trated this,  when  we  were  study- 
ing quadrangles,  by  a  bent  twig, 
or  by  a  bent  lamp-lighter.  If  the 
twig  is  bent  into  the  form  of  a  triangle,  and  its 
ends  held  together,  it  cannot  be  altered  in  'shape. 
But,  if  in  the  form  of  any  other  polygon,  you  can 
flatten  or  stretch  it  into  different  shapes,  which  will 
not  only  be  isoperimetrical,  but  will  have  the  sides 
unchanged. 

4.  When  a  quadrangle  is  put  into  the  largest 


What  is  a  polygon  ?  An  inscribed  polygon  ?  A  polygon  of 
three  sides  ?  Of  four  sides  ?  Of  five  sides  ?  Of  six  sides  ? 
Suppose  that  a  man  measures  how  long  the  sides  of  his  field 
are,  will  that  tell  him  how  large  the  field  is  ?  How  can  you 
Bhow  that  it  will  not,  by  a  bent  twig  ?     But  suppose  his  field 

7# 


78  GEOMETRY. 

form  it  can  have  without  altering  its  sides,  it  can 
be  inscribed  in  a  circle. 

5.  You  can  put  a  circle  about  any  triangle  you 
please ;  but  the  triangle  is  the  only  polygon  that 
can  always  be  inscribed  in  a  circle. 

6.  When  a  quadrangle  is  inscribed  in  a  circle, 
the  sum  of  either  two  opposite  angles  is  equal  to 
two  right  angles.  You  remember  that  I  have 
already  told  you  that  an  angle  with  the  vertex  in 

a  circumference  is  measured 
by  half  the  arc  between  its 
sides.  But  the  arc  between 
the  sides  of  one  angle,  in  this 
quadrangle,  added  to  the  arc 
between  the  sides  of  the  op- 
posite angle,  makes  up  the 
whole  circumference,  and  the  sum  of  the  angles  is 
measured  by  half  the  sum  of  the  arcs  ;  that  is,  by 
half  a  circumference  ;  that  is,  by  one  hundred 
and  eighty  degrees  ;  that  is,  the  sum  of  two  right 
angles. 


has  only  three  sides  ?  Suppose  it  has  four  sides,  what  angles 
must  they  make  to  have  his  field  the  largest  possible  ?  When 
a  quadrangle  is  in  its  largest  form,  what  can  you  say  about 
its  vertices?  Suppose  you  wish  to  lay  seven  sticks  on  the 
floor  so  as  to  enclose  the  most  space  you  can,  how  will  you  lay 
them  ?  Suppose  you  take  seven  other  sticks  more  nearly  equal 
in  length,  but  whose  lengths  added  together  make  just  as 


GEOMETRY.  79 


I 

^m  7.  Any  polygon  whatever^  when  it  is  inscribed 
Hjb  a  circle,  is  in  the  largest  form  that  it  can  ]y^ 

put  without  altering  the  length 

of  the  sides.     If  you  had  some 

sticks  of  different  lengths  that 

you  wished   to  lay  down  as  a 

play-fence  upon  the  ground,  you 

could  make  the  largest  field  of 

them  by  laying  them  in  such  a  position  that  the 

ends  of  the  sticks  shall  all  be  in  the  circumference 

of  one  circle. 

8.  When  two  isoperimetrical   polygons,  of  the 

same  number  of  sides,  are  inscribed  in  circles,  that 

polygon  is  largest  which  has  its  sides  most  nearly 

equal.     Thus, 

if  the  inscribed 

pentagons      A 

and  B  were  iso- 
perimetrical, A 

would   be  the 

larger,  because  its  sides  are  equal,  although  E  would 

be  in  the  larger  circle. 


much  as  the  first  seven,  with  which  set  can  you  enclose  most 
space  on  the  floor  ?  If  five  children  take  hold  of  a  long  loop 
of  string,  each  taking  hold  with  one  hand,  how  must  they 
etand  so  as  to  make  the  opening  in  the  loop  largest  ?  If  a 
sixth  child  comes  in  and  takes  hold,. will  they  make  the  loop 
larger  or  smaller  ?    Suppose,  now,  that  each  child  takes  hold. 


80  GEOMETRY. 

9.  And,  therefore,  of  all  isoperimetrical  poly- 
gons of  the  same  number  of  sides,  that  one  is  the 
largest  which  has  its  sides  exactly  equal,  and 
•which  can  be  inscribed  in  a  circle. 

10.  When  a  polygon  with  equal  sides  can  bt 
inscribed  in  a  circle,  it  is  called  a  regular  polygon. 
Not  only  are  its  sides  equal  to  each  other,  but  its 
angles  also  are  equal  to  ea<jh  other.  An  equilateral 
triangle  is  a  regular  triangle,  and  a  square  is  a 
regular  quadrangle. 

11.  A  regular  polygon  is  larger  than  any  iso- 
perimetrical polygon  of  the  same  number  of  sides. 
This  is  just  what  I  told  you  in  the  ninth  section. 

12.  Of  two  isoperimetrical  regular  polygons, 
that  is  greater  which  has  the  greater  number  of 
sides.  That  is  to  say,  a  square  is  greater  than 
its  isoperimetrical  equilateral  triangle ;  a  regular 
pentagon  greater  than  its  isoperimetrical  square  ; 
and  so  on. 

13.  Suppose  you  wish  to  enclose  some  land  in 
the  middle  of  a  great  field,  with  sixty  panels  of 
fence.  If  the  fence  w^as  put  into  a  square  form, 
fifteen  panels  on  a  side,  it  would  enclose  more  than 
if  put  into  a  triangle  with  twenty  panels  on  a  side ; 

with  both  hands  ?  What  is  meant  by  a  regular  polygon  ?  Two 
isoperimetrical  polygons  ?  What  is  the  largest  of  isoperimetri- 
cal polygons  of  the  same  number  of  sides?  What  is  the 
largest  of  isoperimetrical  regular  polygons?    Of  all  isoperi- 


GEOMETRY.  81 

it  would  enclose  still  more  if  put  round  a  regular 
pentagon,  twelve  panels  on  a  side  ;  still  more  as  a 
hexagon  ten  panels  on  a  side  ;  and  most  of  all  if 
put  in  a  regular  polygon  of  sixty  sides,  one  panel 
on  a  side. 

14.  We  can  fancy  a  circle  to  be  a  regular  poly- 
gon, with  more  sides  than  can  be  counted,  and  each 
side  too  short  to  be  seen.  So  that,  of  all  isoperi- 
metrical  figures,  the  circle  is  the  very  largest. 
Sixty  rods  of  stone  wall  could  not  be  put  into  any 
shape  and  enclose  so  much  land  as  if  it  were  put  in 
a  circle. 


CHAPTER    XXIV. 

HOW  MUCH  FURTHER  IS  IT  ROUND  A  HOOP  THAN 
ACROSS  IT? 

1.  Everybody  knows  that  it  is  about  three 
times  as  far  round  a  circle  as  it  is  across  it.  But, 
if  you  measure  how  far  it  is  across  your  hoop,  you 
will  find  that  a  string  of  three  times  that  length 
will  not  quite  go  around  it. 

metrical  figures  which  is  largest  ?  Suppose  you  wish  to  make 
a  string,  lying  on  the  floor,  enclose  as  much  space  on  the  floor 
as  you  can,  how  will  you  arrange  it  ? 

How  much  further  is  it  round  a  circle  than  across  it  ?  What 


82  GEOMETRY. 

And  now  the  rest  of  this  chapter  will  be  too 
hard  for  children  that  have  not  learned  a  little 
Arithmetic.  If  you  have  not  learned  how  to 
multiply  and  divide,  you  will  have  to  go  to  chap- 
ter XXVI.,  without  understanding  much  about  this 
chapter  and  the  next.  Still,  I  think  you  will  do 
well  to  study  these  chapters,  and  learn  what  you 
can  out  of  them,  even  if  you  do  not  know  how  to 
cipher. 

2.  The  circumference  of  a  circle  and  its  diame- 
ter are  nearly  in  the  same  proportion  as  the  num- 
bers 22  and  7.  So,  if  you  measure  across  the 
hoop,  and  take  a  string  three  and  one  seventh 
times  that  length,  you  will  find  it  just  go  round 
the  hoop. 

3.  If  the  diameter  of  a  circle  is  seven  inches, 
the  circumference  will  lack  only  a  hairbreadth  of 
twenty-two  inches ;  if  the  diameter  is  seven  feet, 
the  circumference  will  not  lack  the  breadth  of  your 
slate-pencil  of  being  twenty-two  feet.  Now  ask 
some  one  to  show  you  a  circle  about  seven  inches 
in  diameter,  such  as  a  breakfast-plate ;  and  a  cir- 


is  the  most  common  and  roughest  answer  to  this  question  ? 
(3  times.)  What  is  a  more  exact  answer?  (22)  How 
nearly  would  this  give  the  circumference  of  a  circle  as  large  as 
a  plate  ?  How  nearly  the  circumference  of  a  large  cistern  ? 
What  still  more  exact  answer  can  you  give  ?     (3'1416.)     Can 


• 


GEOMETRY.  88 


cle  seven  feet  in  diameter,  such  as  a  great  hoop 
that  a  man  can  walk  through  with  his  hat  on. 
You  can  then  judge  how  nearly  the  circumference 
and  diameter  are  in  the  same  proportion  as  tlic 
numbers  twenty-two  and  seven. 

4.  Nobody  knows,  and  nobody  ever  can  know, 
exactly  how  many  times  further  it  is  round  a  circle 
than  across  it.  It  is  very  often  true  in  Geometry 
that  we  cannot  express  by  figures  the  lengths  of 
two  lines.  There  are  no  two  numbers  in  the  same 
proportion  as  the  side  and  diagonal  of  a  square. 
Nobody  can  tell  exactly  how  many  times  longer 
the  diagonal  is  than  the  side  of  the  square.  And 
in  like  manner  there  are  no  'two  numbers  in  the 
same  proportion  to  each  other  as  the  circumfer- 
ence and  diameter  of  a  circle. 

5.  But  we  very  often  want  to  speak  of  this  pro- 
portion, and  we  Avant  to  have  some  short  name  for 
it.  Geometers  have  generally  agreed  to  call  it  pi : 
which  is  the  name  of  the  Greek  letter  iot  p,  and 
it  is  written  tt. 

6.  I  have  told  you  that  tt  is  nearly  twenty-two 


the  answer  be  exactly  given  in  figures  ?  What  Greek  letter  is 
used  to  express  the  exact  proportion  of  a  circumference  to  a 
diameter  ? 

If  there  are  pupils  in  the  class  who  have  studied  arithmetic, 
let  them  answer  such  questions  as  the  following,  using  twenty- 
two  sevenths  in  the  head,  or  3.1416  on  the  slate.   The  simplest 


84  GEOMETRY. 

sevenths.  And,  in  almost  every  question  about 
circles  that  you  will  want  to  answer,  this  is  exact 
enough.  It  is  nearly  enough  exact  for  workmen 
to  work  by  in  making  tinned  ware,  or  anything  in 
which  the  diameter  is  less  than  large  wash-tubs. 

7.  So,  if  you  know  what  the  diameter  of  a  cir- 
cle is,  and  want  to  find  out  how  long  the  circum- 
ference is,  you  must  multiply  the  diameter  by 
twenty-two,  and  divide  the  product  by  seven.  If 
you  know  what  the  circumference  is,  and  want  to 
find  out  what  the  diameter  is,  you  must  multiply 
the  circumference  by  seven,  and  divide  the  product 
by  twenty-two. 

8.  But,  perhaps,  you  will  at  some  time  wish  to 
be  more  exact,  and  then  it  will  be  better  to  use  a 
decimal  fraction.  Perhaps  you  have  not  learned 
decimals  yet ;  but,  after  you  have  learned  them, 
you  may  want  to  use  a  better  value  of  rr,  and  you 
can  then  turn  back*  to  this  page,  and  find  that  tt  is 
very  nearly  equal  to  3-1416. 


and  best  way  of  reading  decimal  fractions  is  simply  to  say 
*'  decimal  one  four  one  six." 

What  is  the  diameter  of  a  hoop  44  inches  in  circumference  ? 
What  is  the  circumference  of  a  hoop  21  inches  in  diameter  ? 
14?  3-5?  10-5?  17-5?  11? 


<i>EOMETBy.  85 

CHAPTER    XXV. 

HOW   TO   MEASURE   THE   SIZE   OF   A    CIRCLE. 

1.  I  HAVE  told  you  that  men  measure  surfaces 
by  squares.  They  find  out,  if  they  can,  how  many 
squares  it  would  take  to  cover  the  surfaces,  if  the 
side  of  each  square  was  just  one  inch,  or  one  foot, 
or  one  yard.  I  have  told  you,  also,  that  we  can 
easily  find  out  how  many  such  squares  it  takes  to 
cover  a  large  square ;  that  is,  we  can  find  the 
measure  of  a  square  by  multiplying  the  number 
of  inches,  feet  or  yards,  on  a  side,  by  itself;  that 
is,  by  the  same  number. 

2.  Now,  if  you  draw  a  square 
with  its  sides  tangent  to  a  circle, 
the  sides  of  this  square  will  be 
each  equal  to  the  diameter  of  the 
circle.  The  measure  of  such  a 
square  is  found,  then,  by  multi- 
plying the  length  of  the  diame- 
ter by  itself  And  the  measure  of  the  circle  can 
be  found  by  multiplying  the  measure  of  the  square 
by  one  quarter  of  tc. 


If  the  class  have  not  studied  any  arithmetic,  this  chapter 
must  be  omitted  until  a  review. 

How  do  men  measure  surfaces  ?  How  do  you  find  the  meas- 
ure  of  a  square  ?    How  do  you  measure  a  circle  ?    Let  the 

8 


86  GEOMETRY. 

3.  Since  n  is  a  little  more  than  three,  the  circle 
is  a  little  more  than  three  quarters  of  the  square. 
If,  for  example,  the  diameter  of  a  circle  is  six 
inches,  the  square  that  will  just  enclose  it  contains 
six  times  six,  or  thirty-six  square  inches  ;  and  the 
circle  contains  a  little  more  than  three  quarters  of 
this.  Three  quarters  of  thirty-six  is  twenty-seven  ; 
and,  adding  a  little  more,  would  make  it  about 
twenty-eight  inches. 

4.  If  you  wish  to  be  more  exact,  you  must  mul- 
tiply the  thirty-six  square  inches  by  one  quarter 
of  twenty-two  sevenths :  that  is,  by  eleven  four- 
teenths. Or,  in  other  words,  we  must  multiply 
thirty-six  by  eleven,  and  divide  by  fourteen,  w^hich 
w'ill  give  us  about  twenty-eight  and  one  half 
square  inches  for  the  size  of  a  circle  six  inches  in 
diameter. 

5.  And,  to  be  very  exact  in  finding  the  size 
of  a  circle,  you  must  multiply  the  diameter  by 
itself,  and  then  by  the  decimal  '7854,  which  is  one 
quarter  of  3*1416. 

6.  Circles  are  larger  or  smaller  in  the  same 
proportion  as  the  squares  built  on  their  diameters. 


teacher  draw  a  square,  and  ask,  What  figure  is  this  ?  In- 
scribe a  circle,  and  ask.  What  figure  is  this  ?  How  large  a 
part  of  the  square  does  it  enclose?  (|).  More  exactly? 
(JLi)     Still  more  exactly?     (-7854.)     By  what  part  of  n 


OBOMETRY.  87 

And  something  like  this  is  true  of  all  sorts  of 
surfaces.  Two  similar  surfaces  are  always  in  pro- 
portion to  the  squares  of  similar  lines  in  those 
surfaces.  If  we  have  two  polygons  of  the  same 
shape,  they  are  of  a  size  proportioned  to  the 
squares  on  their  corresponding  sides  or  diagonals. 
If  a  side  in  one  is  twice  as  long  as  a  corresponding 
side  in  the  other,  then  one  polygon  is  four  times 
the  size  of  the  other,  because  twice  two  are  four. 
If  one  side  were  three  times  as  long  as  the  corre- 
sponding side  in  the  other  polygon,  one  polygon 
would  be  nine  times  as  large  as  the  other,  because 
three  times  three  are  nine. 

7.  I  am  so  anxious  that  you  should  remember 
this,  that  I  will  tell  it  to  you  again  in  other  words. 
All  similar  surfaces  are  in  proportion  to  the 
squares  of  corresponding  lines ;  so  that  we  may 
find  the  proportion  between  the  surfaces  by  multi- 
plying the  number  that  expresses  the  proportion 
between  the  lines  by  itself. 

Suppose  two  dogs  were  of  exactly  the  same 
shape,  but  that  one  was  twice  as  high  as  the  other. 
Then  its  tail  would  be  twice  as  long  as  the  other's, 


must  you  multiply  the  square  in  order  to  find  the  measure 
of  the  circle  ?  How  many  square  inches  in  a  circle  five  inches 
in  diameter.  Suppose  a  little  man,  just  one  foot  high,  and  a 
man  six  teet  high,  how  much  more  cloth  will  it  take  to  clothe 


88  OEOMETK^. 

its  ears  would  be  twice  as  long,  its  eyes  twice  as 
wide  apart :  and  whatever  line  you  chose  to  meas- 
ure in  one,  it  would  be  twice  as  long  as  the  same 
line  in  the  other.  But  its  skin  would  be  four 
times  as  large,  the  surfa,ce  of  its  eye  would  be  four 
times  as  large,  it  would  take  four  times  as  much 
leather  to  make  boots  for  it,  or  four  times  as  much 
lather  to  shave  it ;  that  is,  whatever  surface  you 
measured  on  the  one  dog,  you  would  find  it  four 
times  as  large  as  the  same  surface  on  the  other. 

Suppose  that  we  had  a 
foot-ball  ten  inches  in  diam- 
eter, and  a  little  batting- 
ball  two  inches  in  diameter. 
The  diameter  of  the  foot-ball 
would  be  five  times  as  much 
as  that  of  the  batting-ball,  and  it  would  take 
twenty-five  times  as  much  leather  to  cover  it, 
because  five  times  five  is  twenty-five. 


one  than  the  other?    How  much  more  yam  to  knit  his  stock- 
ings? 


GEOMETRY.  89 

CHAPTER    XXVI. 

ABOUT   CURVATURE. 

1.  Suppose  that  our  boy,  wheeling  his  barrow 
over  the  light  fallen  snow,  went  winding  about  the 
field,    making   a    curved   track,  ^--n,^ 
which    curved   in    some    places  /\        ^v,_?y 
more   than   in   others.      Let   us  V 

,  suppose  that  he  began  as  though  he  were  going  to 
make  a  large  circle,  but  kept  turning  shorter  and 
shorter,  and  ended  when  he  was  turning,  as  though 
he  would  make  a  very  little  circle.  Then  we 
should  say  that  his  track  had,  at  first,  a  large 
radius  of  curvature,  but  at  the  end  had  a  small 
radius  of  curvature. 

2.  Let  us  suppose  that  the  boy  was  tied,  by  a 
long  rope,  to  the  trunk  of  a  large  tree ;  and  that, 
as  he  went  round  and  round  the  tree,  the  rope 
wound  up  upon  the  tree-trunk,  shorter  and  shorter, 
and  drew  the  boy  nearer  and  nearer  to  the  tree. 
Then  the  rope  would  be  the  radius  of  curvature  of 
the  boy's  path. 

3.  Hold  a  spool  of  thread  still,  on  your  slate, 
and  let  it  be  the   trunk  of  the  tree.     Then  tie 


"What  is  the  name  of  a  curve  that  bends  equally  in  every 
part?  How  would  you  draw  such  a  curve  upon  the  black- 
board?    If  I  unwrap  a  thread  from  a  spool,  holding  the  spool 

8* 


90  GEOMETRY. 

the  end  of  your  slate-pencil  to 
the  end  of  the  thread,  and,  by 
keeping  the  thread  tight  as  you 
unwind  it,  you  may  draw  a 
track  like  that  of  the  boy's 
wheelbarrow.  The  thread  that  is  unwound  will 
be  the  radius  of  curvature  of  this  mark.  The 
radius  of  curvature  will  be  very  short  where 
the  pencil  is  close  to  the  spool,  and  grow  longer 
as  you  unwrap  the  thread.  It  will  be  different 
for  every  point  in  the  curve ;  because  you  can- 
not move  the  pencil  without  either  winding,  or 
else  unwinding,  the  thread. 

4.  We  call  this  thread  the  radius  of  curvature, 
because  it  is  to  the  curve  like  a  radius  to  the  cir- 
cle. We  call  it  the  radius  of  curvature^  because 
it  shows  us  how  much  a  curve  curves  or  bends. 
When  the  radius  of  curvalrure  is  short,  the  curve 
bends  very  much  ;  and  when  the  radius  of  curva- 
ture is  long,  the  curve  bends  less;  and  so  the 
radius  of  curvature  measures  the  bending  or  curv- 
ature of  the  curve. 

5.  If  we  draw  a  circle,  with  its  centre  at  the 
point  where  the  thread  is  just  leaving  the  spool. 


still,  and  keeping  the  thread  tight,  what  sort  of  a  curve  shaU  I 
draw?  What  relation  will  the  circumference  of  the  spool 
have  to  this  curve  ?    What  shall  we  call  the  straight  part  of 


GEOMETRY.  •        91 

that  is,  where  the  thread  is  tangent  to  the  spool, 
and  make  the  radius  of  the  circle  just  equal  to  the 
thread  that  has  been  unwound,  that  is,  equal  to 
the  radius  of  curvature,  then  that  circle  will  ex- 
actly fit  the  curve  at  the  point  which  the  slate- 
pencil  is  then  marking.  So  that  the  radius  of 
curvature,  at  any  point  of  a  curve,  is  the  radius 
of  the  circle  that  will  exactly  fit  the  curve  at  that 
point. 

6.  Every  curve  can  be  imagined  as  made  in  a 
similar  way,  by  unwrapping  a  string  off  from  some 
other  curve  ;  and  this  other  curve  is  called  the 
evolute  of  the  first  curve. 

7.  But  the  evolute  of  a  circle  is  a  point ;  be- 
cause the  string  that  makes  a  circumference  must 
neither  wind  up  nor  unwind. 

8.  The  evolute  of  the  boy's  track  is  the  circum- 
ference of  the  trunk  of  the  tree ;  and  the  evolute 
of  the  pencil-mark  is  the  circumference  of  the 
spool. 

9.  You  may  drive  a  row  of  pins  into  a  soft  pine 
board,  making  the  row  curved.     Then  tie  one  end 

the  thread  which  runs  between  my  hand  and  the  spool  ?  What 
does  the  radius  of  curvature  measure  ?  To  what  circle  is  the 
radius  of  curvature  a  radius  ?  How  do  we  imagine  all  curves 
drawn  ?  What  is  the  evolute  of  a  circle  ?  Let  the  teacher 
provide  the  board  and  pins  to  -show  the  illustration  of  sec- 
tion nine. 


92 


GEOMETRY. 


of  a  thread  to  the  foot  of  the  last  pin,  and  the 
other  end  of  the  thread  to  a  lead-pencil  near  its 
point.  By  keeping  the  string  stretched,  and  sweep- 
ing it  round  so  as  to  wrap  up  and  unwrap  upon 
the  fence  of  pins,  you  may  draw  a  curve  whose 
evolute  will  be  the  row  of  pins.  This  pencil-mark, 
you  will  easily  see,  is  made  of  little  arcs  of  cir- 
cles, whose  centres  are  the  pins,  and  the  length 
of  thread  from  a  pin  to  its  little  arc  is  the  radius 
of  curvature  at  that  place. 


CHAPTER    XXVII. 

ABOUT   A    WHEEL   ROLLING. 

1.  When  a  wagon  is  going  upon  a  straight  and 
level  road,  look  at  the  head  of  a  spike  in  the  tire 
of  one  of  the  w^heels,  and  you  will  see  that  it 


moves  in  beautiful  curves,  making  a  row  of  arches 
that  is  called  a  cycloid. 


Let  the  teacher  take  a  tin  cup,  a  ribbon-block,  or  something 
of  the  kind,  and  roll  it  carefully  along  the  bottom  of  the  black- 
board, watching  and  marking  with  chalk  the  path  of  a  spot 


GEOMETRY.  93 

2.  That  is  to  say,  a  cycloid  is  the  path  of  a 
point  in  the  circumference  of  a  circle  rolling  on  a 
straight  line.  You  can  draw  part  of  a  cycloid  by 
putting  the  point  of  your  pencil  into  a  little  notch 
in  the  edge  of  a  spool,  and  tying  it  fast,  so  that 
the  point  of  the  pencil  shall  be  kept  just  at  the 
edge  of  the  spool ;  and  then  rolling  the  spool  care- 
fully and  slowly  against  the  inside  of  the  frame  of 
the  slate, 

3.  You  will  see,  I  think,  that  each  arch  in  the 
cycloid  must  be  just  as  high  from  c  to  D  as  the 
diameter  of  the  circle  that  makes  it ;  and  just  as 
wide  at  the  bottom,  from  A  to  B,  as  the  whole  cir- 
cumference of  the  circle. 

4.  But  you  will  have  to  study  Geometry  a  good 
while,  before  you  can  prove  the  other  interesting 
things  which  I  am  going  to  tell  you.  You  can 
easily  understand  what  I  am  going  to  tell  you ; 
but  you  cannot  understand  how  I  know  it,  as  you 
can  what  I  told  you  in  the  last  section. 

5.  The  length  of  the  curve,  A  d  b,  in  each  arch 


on  the  side.  Then  ask,  What  is  the  name  of  this  curve  ?  What 
is  the  height  of  the  arch,  compared  with  the  cup^  with  which 
I  drew  it  ?  What  is  the  breadth  of  the  arch  at  the  bottom  ? 
What  is  the  length  of  the  curve  of  the  arch  ?  What  is  the 
space  inclosed  between  the  arch  and  the  bottom  of  the  board  ? 
(still  comparing  with  the  cup.)  Let  the  teacher  inscribe  a 
circle,  of  the  size  of  the  cup,  and  ask,  Are  these  horns  larger 


94 


GEOMETRY. 


of  a  cycloid,  is  just  four  times  the  height  of  the 
arch  ;  that  is,  four  times  the  diameter  of  the  circle 
that  made  the  cycloid. 

6.  The  whole  space  that  is  enclosed  between  the 
arch  of  the  cycloid  and  the  straight  line  on  which 

it  stands,  is  just 
three  times  as 
large  as  the  cir- 
cle that  made  the 
cycloid.  So,  when 
a  circle  is  in- 
scribed between 
the  arch  and  the  line,  the  curious  three-cornered 
figures  on  each  side  of  the  circle  are  each  exactly 
as  large  as  the  circle  itself. 

7.  Now,  if  you  have  studied  Arithmetic,  you 
will  understand  that,  if  a  wheel  is  three  feet  in 
diameter,  the  head  of  a  spike  in  the  tire  travels 
just  twelve  feet  from  where  it  leaves  the  ground 
until  it  touches  the  ground  again.  The  spots  where 
it  touches  the  earth  will  be  nine  feet  and  three  sev- 
enths of  a  foot  apart.     And  the  space  between  its 


or  smaller  than  the  circle  ?  Suppose  that  your  hoop  has  a 
spot  on  one  side  of  it,  in  what  curve  will  the  spot  move  when 
the  hoop  is  rolling  straight  forward  ?  How  high  will  it  go 
from  the  ground  ?  (Diameter  of  hoop.)  How  far  apart  will 
the  places  be  where  it  comes  to  the  ground  ?    How  far  will  the 


I 


GEOMETRY. 


95 


path  and  the  ground  will  be  three  times  eleven 
fourteenths  of  nine  square  feet ;  that  is,  twenty- 
one  square  feet  and  three  fourteenths  of  a  square 
foot. 


CHAPTER    XXVIII. 

MORE   ABOUT   A   ROLLING   WHEEL. 

1.  The  head  of  a  spike  in  the  tire  of  a  rolling 
wheel  is  moving,  at  each  instant,  at  right  angles 
to  a  line  joining  it  to  the  bottom  of  the  wheel. 

2.  That  is  to  say,  if  a  straight  line  is  drawn 
from  the  bottom  of  the  rolling  wheel  to  the  head 
of  the  spike,  and  if  a  tangent  to  the  cycloid  is 
drawn  through  the  head  of  the  spike,  this  straight 
line  will  be  at  right  angles  to  this  tangent. 

3.  And  this  straight  line  is  exactly  half  of  the 
radius  of  curva- 
ture of  that  point 
in  .the  cycloid.  So 
that,  at  the  top  of 
an  arch  of  the  cy- 
cloid the  radius 
of  curvature  will 


spot  travel  in  going  from  one  place  to  the  next  ?     (Four  times 
diameter  of  hoop.) 

Which  of  you  can  tell  me  what  a  cycloid  is  ?    If  you  draw  a 
line  at  right  angles  to  a  cycloid,  where  will  it  pass  ?  (Through 


96 


aEOMETRY. 


be  twice  the  diameter  of  the  circle,  and  as  you  go 
down  the  arch  the  radius  of  curvature  will  be 
shorter  and  shorter,  until  just  at  the  foot  of  the 
arch  the  radius  of  curvature  will  be  of  no  length 
at  all. 

4.  The  evolute  of  a  cycloid  is  a  cycloid  of  ex- 
actly the  same  size.     That  is  to  say,  if  we  should 

fasten  a  string  in 
the  point  between 
two  arches  of  a 
cycloid,  just  long 
enough  to  wrap  on  the  curve  up  to  the  middle  of 
the  arches,  its  end,  as  it  wrapped  and  unwrapped, 
would  move  in  a  cycloid  exactly  like  that  to  which 
it  was  fastened. 

5.  If  a  cycloid  be 
turned  upside  down,  and 
we  fancy  the  inside  of 
it  to  be  very  exceedingly 
slippery,  then  there  are 
two  curious  things  about  it. 


the  point  where  the  circle  making  the  cycloid  touched  the  line 
on  which  it  rolled  when  making  that  place  in  the  cycloid.)  The 
radius  of  curvature  is  at  right  angles  to  a  curve  —  what  part  of 
the  radius  of  a  cycloid  is  cut  off  by  a  straight  line  joining  the  feet 
of  the  arch  ?  (One  half)  How  long  is  the  radius  of  the  cycloid 
at  the  top  of  the  arch  ?  How  long  at  the  bottom  ?  What  is 
the  evolute  of  a  cycloid  ?    Explain  what  you  mean  by  this  ? 


I 


GEOMETRY.  97 

If  I  want  to  slide  anything  from  A  down  to  B, 
there  is  no  curve,  nor  straight  line,  down  which  a 
thing  would  slide  so  quickly  as  down  the  cycloid. 
If  a  hill  was  hollowed  out  in  that  shape,  sleds 
would  run  down  it  faster  than  they  could  down 
any  other  shaped  hill  of  the  same  height  and  the 
same  breadth  at  the  bottom. 

6.  The  second  curious  thing  about  •  sliding  on 
the  inside  of  a  cycloid  is,  that  it  takes  always 
exactly  the  same  time  to  slide  to  the  bottom,  how- 
ever high  up  or  low  down  you  start.  If  A,  in  the 
last  figure,  is  the  top  of  such  a  hill,  and  c  the 
lowest  point,  it  will  take  a  sled  exactly  as  long  to 
go  from  B  to  c,  as  to  go  from  A  to  c.  But  this, 
you  must  remember,  is  only  when  we  imagine  the 
hill  and  the  runners  of  the  sleds  to  be,  both  of 
them,  perfectly  slippery ;  so  that  there  shall  be  no 
rubbing.  In  that  case,  if  the  road  from  A  to  c 
was  two  miles  long,  it  would  only  take  a  sled 
twenty-eight  seconds  to  come  down  the  whole 
length.     And,  if  it  starts  from  any  other  place  on 


Suppose  tAvo  wires  going  from  the  north-east  corner  of  the 
ceiling  to  the  south-west  corner  of  the  floor,  one  wire  straight, 
the  other  a  part  of  a  cycloid,  down  which  wire  would  anything 
slide  the  more  quickly?  Suppose  one  wire  went  from  the 
north-east  corner  of  the  ceiling  to  the  south-west  corner  of  the 
ceiling,  hanging  down  in  the  formof  a  whole  arch  of  a  cycloid, 
how  mucli  longer  would  it  take  anything  to  slide  from  the  ceil-^ 

9 


98  aEOMETRY. 

the  road,  say  from  b,  it  Avill  still  take  twenty- 
eight  seconds  to  get  to  c. 

7.  If  the  road  from  a  to  c  is  half  a  mile  long, 
a  sled  will  come  down  in  fourteen  seconds. 

8.  If  a  board  is  sawed  out  in  the  form  of  a 
cycloid,  and  a  little  gutter  made  on  the  inside  of 
the  curve,  you  can  try  this  by  holding  two  mar- 
bles, say  one  at  A  and  the  other  at  d,  and  letting 
go  of  them  at  the  same  instant. .  They  will  meet 
exactly  at  c,  one  coming  the  whole  way  A  c,  while 
the  other  is  coming  the  short  distance  D  c. 


CHAPTER    XXIX. 

WHEELS   ROLLING   ROUND   A    WHEEL. 

1.  When  one  circle  rolls  around 
another,  instead  of  rolling  on  a 
straight  line,  any  point  iii  the  cir- 
cumference of  the  rolling  circle 
travels  in  a  curve  called  an  epicy- 
cloid.^  You  can  draw  an  epicycloid 


ing  to  the  lowest  part  of  the  wire,  than  it  would  take  for  it  to 
slide  from  a  place  one  foot  from  the  middle?  (No  longer.) 
The  teacher  should  endeaTor  to  obtain  (from  the  prudential 
-committee)  the  board  described  in  section  eight. 

*  EpisS'clold. 


GEOMETRY.  ^ 

by  rolling  carefully  the  spool  (with  a  pencil  tied 
to  it)  around  some  round  thing  held  still  on  your 
slate. 

2.  Set  a  lamp  on  a  table  in  one  corner  of  the 
room,  and,  in  the  farthest  corner  of  the  room,  on  a 
table  of  nearly  the  same  height,  set  a  bright  tin 
cup,  or  a  glass  tumbler,  nearly  full  of  milk.  On 
a^  surface  of  the  milk  you  will  see  a  bright  curve 
shaped  like  the  inner  line  in  this  ^.--^ 
figure.  It  is  an  epicycloid ;  such 
as  would  be  made  by  a  circle  of 
one  quarter  the  diameter  of  the 
cup  rolling  on  a  circle  half  the 
size  of  the  cup.  You  can  make 
it  by  daylight,  by  setting  the  cup  of  milk  in  the 
sunshine,  early  in  the  morning  or  late  in  the  after- 
noon. 

3.  Epicycloids  will  be  of  different  shapes,  accord- 
ing to  the  proportion  which  the  two  circles  bear  to 
each  other.  The  smaller  the  rolling  circle  is  in 
proportion  to  the  other,  the  more  nearly  will  an 
arch  of  the  epicycloid  be  like  an  arch  of  the 
cycloid. 

What  is  an  epicycloid  ?  How  does  it  differ  from  a  cycloid  ? 
Let  the  teacher  draw  an  epicycloid  as  directed  in  section  one, 
and  teach  the  children  to  do  so.  Have  any  of  you  seen  the 
cow's  foot  in  a  cup  of  milk  ?  What  is  the  geometrical  name 
of  this  curve  ?    What  must  be  the  proportion  between  the  cix- 


100  GEOMETRY. 

4.  The  evolute  of  an  epicycloid  is  a  smaller 
epicycloid  of  the  same  shape :  and  the  evolute  of 
that  evolute  must  be  a  still  smaller  epicycloid.  So 
that  we  may  fancy  epicycloids  packed  one  within 
another  like  pill-boxes. 

5.  The  epicycloid  of  section  second  is  sometimes 
called  by  children  the  cow's  foot  in  a  cup  of  milk. 

The  figure  in  the  margin  reprog- 
sents  this  epicycloid  with  its  nest 
of  evolutes  packed  one  within 
the  other.  If  a  string  is  fast- 
ened at  the  point  where  the 
arches  of  the  epicycloid  come 
together,  and  is  just  long  enough  to  wrap  round  to 
the  middle  of  the  arch,  then,  as  it  unwraps,  the 
end  will  move  in  a  larger  epicycloid  of  exactly  the 
same  shape. 

6.  When  the  circles  are  of  the 
same  size,  the  epicycloid  will  have 
but  one  arch.  The  ends  of  the  arch 
will  come  together  at  the  same  point. 
The  figure  in  the  margin  will  show 
the  shape  of  this  epicycloid  and  its  evolutes. 


cles  to  make  this  epicycloid  ?  What  is  the  evolute  of  any  epi- 
cycloid ?  When  the  circles  are  of  the  same  size,  "what  will  be 
the  shape  of  the  epicycloid  ? 


GEOMETRY.  101 

CHAPTER    XXX* 

OF    A   WHEEL    ROLLINa     ON     THE    INSIDE     OF    A 
HOOP. 

1.  When  a  circle  rolls  on  the 
inside  of  another  circle,  instead 
of  on  the  outside,  the  curve  is 
called  a  hypocycloid.^ 

2.  Suppose  your  slate-pencil*  were  ; s^x flight,  ^o 
that  it  would  lie  flat  on  .the  slate,  aiicl  make' a 'mark 
as  broad  as  the  pencil  is  long'     Then  suppose,  you 

were  to  put  your  pencil  across 
one  corner  of  your  slate  like  a 
hypotenuse,  and  slide  first  one 
end  up  to  the  corner,  and  then 
the  other,  keeping  both  ends  all 
the  time  touching  the  slate- 
■  frame.  You  would  make  a 
white  mark  in  the  corner,  of  a  curved  three- 
cornered  shape,  like  this  figure.  The  curve  inside 
would  be  a  hypocycloid. 

3.  If  you  take  the  corner  of  your  slate  for  a 

What  is  a  hypocycloid  ?  Suppose  I  were  to  draw  a  hundred 
right  angles,  putting  the  vertices  of  the  right  angles  together, 
one  exactly  on  another,  making  the  hypotenuses  of  equal 
length,  but  having  no  two  of  them  make  the  same  angle  with 
the  legs,  to  what  kind  of  a  hypocycloid  would  all  these  hypote- 

*  Hiposi'cloid. 

9=^ 


102  GEOMETRY- 

centre,  and  the  length  of  your  pencil  for  a  radius, 
and  draw  a  quarter  of  a  circle,  as  I  have  done  in 
the  last  figure ;  if  you  then  roll  on  the  inside  of 
this  arc  a  circle  whose  diameter  is  one  half  the 
length  of  the  pencil,  it  will  make  the  same  hypo- 
cycloid.  I  have  also  drawn  this  circle  in  the 
figure. 

4.  You  can  draw  a  hypocycloid  by  rolling  the 
spool  and  the  pencil  on  the  inside  of  any  little 

.  hoop  hold  firnily  on  the  slate.  The  rim  of  the 
ci^vef  of  a.  large  wooden  pill-box  will  make  a  nice^ 
little  hoop  for  this  purpose. 

5.  The  evolute  of  a  hypocycloid  is  a  larger  hy- 
pocycloid of  the  same  shape  on  the  outside  of  it ; 
and  the  hypocycloid  itself  may  be  fancied  as  the 
evolute  of  a  smaller  hypocycloid  within  it ;  so  that 
hypocycloids,  like  epicycloids,  are  packed  one 
within  the  other,  like  nests  of  tubs  or  boxes. 

6.  The  hypocycloid  that  is 
made  when  the  diameter  of  the 
rolling  circle  is  one  quarter  of 
the  diameter  of  the  circle  that 
it  rolls  in,  can  be  made  by 
sliding   the   hypotenuse   back- 

nuses  be  tangent  ?  —  that  is,  what  is  the  proportion  between 
the  radii  of  the  two  circles  ?  Suppose  a  man  draws  the  foot 
of  a  ladder  away  from  the  side  of  a  house,  letting  the  ladder 
slip  down  the  side  of  the  house,  to  what  curve  in  the  air  will 


GEOMETRY.  103 

wards  and  forwards  on  the  legs  of  a  riglit  triangle. 
The  hypotenuse  must  be  kept  of  the  same  length, 
and  it  will  always  be  a  tangent  to  the  hypocycloid. 

7.  This  hypocycloid  may  be  called 
a  hypocycloid  of  four  arches;  be- 
cause, as  you  may  see  in  the  figure, 
both  it  and  its  evolutes  have  each 
four  arches. 

8.  If  the  diameter  of  the  spool  is  nearly  half 
that  of  the  hoop,  the  pencil  will  move  across  the 
hoop  in  a  very  flat  curve,  almost  like  a  diameter 
of  the  hoop :  and  the  evolute  at  the  end^  of  the 
curve  will  be  almost  like  two  parallel  straight  lines 
at  right  angles  to  the  end  of  the  diameter  ;  so  that 
the  string  unwrapping  from  the  evolute  will  be 
very  long.  When  the  diameter  of  the  spool  is 
exactly  half  that 
of  the  hoop,  the 
hypocycloid  is  a 
straight  line ;  and 
the  evolute  of  it, 
if  you  can  fancy 
that  there  is  any 

the  ladder  be  all  the  time  a  tangent  ?  If  the  diameter  of  the 
rolling  circle  is  one  fifth  that  of  the  other  circle,  how  many 
arches  will  the  hypocycloid  have?  If  one  fourth?  If  one 
third?  But  what  does  the  hypocycloid  become  when  the 
diameter  of  the  rolling  circle  is  one  half  that  of  the  other  ? 


104  GEOMETRY. 

evolute,  is*  two  parallel  straight  lines  at  right 
angles  to  its  ends. 

9.  If  the  diameter  of  the  spool  is  more  than 
half  that  of  the  hoop,  it  will  make  a  hypocycloid 
like  that  made  by  a  smaller  spool.  If  you  have 
two  spools,  one  of  them  as  much  wider  than  the 
radius  of  the  hoop  as  the  other  is  smaller,  so  that 
the  hoop  w^ill  just  let  the  two  spools  stand  in  it 
side  by  side,  then  one  spool  will  make  exactly  the 
same  hypocycloid  as  the  other. 

10.  These  two  spools  cannot,  of  course,  be  both 
rolling  in  the  hoop  at  the  same  time  ;  but  we  can 
easily  imagine  two  circles  of  the  same  size  as  the 
spools  rolling  in  a  circle  as  large  as  the  hoop. 
Start  the  circles  from  the  posi- 
tion in  which  I  have  drawn 
them  to  rolling  in  opposite  di- 
rections, and  if  you  roll  the 
little  circle  faster  than  the 
large  one,  so  as  to  make  them 
get  round  the  hoop  in  the  same  time,  the  points  in 
the  two  circles  which  are  now  touching  will  keep 
together  all  the  time,  making  the  same  hypocy- 
cloid. 

How  do  the  evolutes  of  a  hypocycloid  differ  from  those  of  an 
epicycloid  ?  In  what  respect  are  they  like  them  ?  What  must 
be  the  size  of  two  spools  that  they  may  make  the  same  hypo- 
cycloid in  the  same  hoop  ? 


GEOMETRY. 


105 


CHAPTER    XXXI. 


ABOUT  A  HANGING   CHAIN. 

1.  When  a  chain 
hangs  from  two  points 
not  directly  under  each 
other,  it  makes  a  beau- 
tiful curve  called  a 
catenary.^  You  must' 
remember  that,  in  order  to  have  a  perfect  catenary, 
we  must  take  a  very  fine  chain,  and  then  take  only 
the  middle  line  in  it. 

2.  Suppose  we  had  four  straight  sticks  joined 
together  by  the  ends,  so  as  to  have  a  sort  of  chain 
of  four  links.  Suppose  the  middle  two  were  equal 
in  length,  and  also  that  the  end  ones  were  equal  to 
each  other.  Hang  them  by  two 
pins  on  a  level,  as  you  see  them  in 
the  •  figure,  and  notice  exactly  in 
what  shape  they  hang.  Now  turn 
them  upside  down,  keeping  them  in 
the  same  shape,  as  you  see  them  in  the  next  fig- 
ure.   They  will  exactly  balance  and  stand  like  the 


What  is  the  geometrical  name  of  the  curve  made  by  a  hang- 
ing chain  ?  If  the  vertices  of  a  polygon  were  perfectly  limber 
hinges,  but  the  sides  stiff,  how  should  we  place  them  to  make 

*  Cat'^n^iry. 


106  GEOMETRY. 

rafters  of  a  double-pitched  roof.  If  you  set  the 
sides  more  nearly  perpendicular,  the  top  will  fall 
in  ;  crowding  the  sides  apart  until 
the  point  of  the  top  gets  lower  than 
the  top  of  the 
sides,  and  then 
pulling  the  sides 
together  again  till  they  touch 
at  the  top,  and  the  two  top  pieces 
hang  straight  down  in  the  middle.  But  if,  on  the 
other  hand,  you  lean  the  sides  together  more  than 
they  should  be,  they  will  fall  together,  crowding 
the  top  up,  until  the  ends  of  the  sides  meet,  and 
the  top  pieces  stand  straight  up,  or  fall  to  one 
side  together.  The  four  sticks,  hinged  together 
at  the  ends,  will  not  stand,  like  an  arch,  unless 
they  make  the  same  angles  with  each  other  as 
they  did  when  they  were  hanging  like  a  chain. 

3.  And  if  we  had  a 
chain  made  of  a  great 
many  short,  stiff  pieces 
of  wood  or  metal,  hinged 
together  by  rivets,  like 
the  little  chain  inside  a 


them  stand  as  an  arch?  Did  you  ever  see  a  gambrel  roof? 
Did  you  like  the  looks  of  it  ?  What  shape  do  you  think  a 
gambrel  roof  should  have  to  look  well  ?     (That  in  which  the 


GEOMETRY. 


1«T 


watch,  we  could  make  it  stand  up  like  an  arch,  if 

we  could  put  it  exactly  in  the  same  form  as  it 

hung;    that   is,  in   a 

catenary  upside  down. 

If  we  arch  it  up  too 

steep  and  pointed,  the 

sides  will  fall  in ;   if 

we  arch  it  too  flat,  the 

top  will  fall  in.     But 

arch  it  exactly  as  it  hung,  and  it  will  stand. 

4.  If  we  fasten  one  end 
of  a  chain  to  a  post,  and 
hang  the  other  end  by  a 
thread  from  the  top  of  a 
higher  post,  the  weight 
of  the  chain  will  pull  the 

thread  inward,  as  in  this  figure. 

But  suppose  another  thread,  tied  to  the  end  of 

the  chain,  should  pass  over  a  little  wheel,  on  a 

level  with  the  end  of  a  chain,  as  at  c,  having  a 

piece  of  the  same  kind  of  chain  hung  to  it  at  b. 

Then  you  can  easily  see  that  the   weight  of  b 

would  pull  A  out  flatter,  and  make  the  thread  hang 

more  nearly  straight  down  by  the  side  of  the  post ; 


rafters  would  hang  if  inverted. )  If  I  draw  a  catenary  on  the 
blackboard,  and  tell  you  how  long  the  radius  is  at  the  bottom, 
can  you  show  me  how  to  find  the  radius  at  any  other  part  of 


lOS  GEOMETRY. 

or,  if  B  were  long  enough,  and  thus  heavy  enough, 
it  would  even  draw  the  thread  outward  toward  c. 

5.  If  the  piece  of  chain  marked  b  is  just  long 
enough  to  pull  the  end  of  A  exactly  under  the  top 
of  the  higher  post,  so  as  to  make  the  thread  hang 
exactly  straight  down,  then  b  will  be  just  as  long 
as  the  radius  of  curvature  of  the  catenary  A  at  its 
lowest  point. 

6.  Let  A,  in  the 
next  figure,  be  the 
lowest  point  of  any 
catenary,  and  c  any 
other  point  in  it 
you  please. 

Draw  a  horizontal  line,  E  d,  making  the  distance 
A  E  equal  to  the  radius  of  curvature  at  A.  Now 
draw  c  B  at  right  angles  to  the  catenary  at  the 
point  c,  and  c  D  will  be  exactly  the  same  length  as 
the  radius  of  curvature  at  c.  Draw  A  F  parallel 
to  c  D,  and  the  straight  line  E  F  will  be  just  as 
long  as  the  piece  of  chain  A  c. 

the  chain?    Can  you  show  me  how  to  find  a  straight  line 
equal  to  any  part  of  the  catenary  ? 


GEOMETRY.  109 

CHAPTER    XXXII. 

THE   PATH   OF   A   STONE   IN   THE   AIR. 

1.  When  a  boy  tosses  up  his  ball  in  the  air, 
the  centre  of  the  ball  moves  in  a  curve  called  a 
parabola.  If  you  toss  up  the  ball  on  the  west 
side  of  the  house  when  the  sun  is  setting,  the 
shadow  against  the  side  of  the  house  will  also 
move  in  a  parabola. 

2.  If  you  hold  a  round  ball  in  such  a  position 
that  its  upper  edge  is  just  as  high  above  the  table 
as  the  blaze  of  a  lamp  is,  then  the  edge  of  the 


shadow  on  the  table  will  be  a  parabola.  A  dinner- 
plate  will  also  make  a  parabola  in  the  same 
manner. 

3.  But  remember  that,  to  be  an  exact  parabola, 
the  ball  must  be  perfectly  it)und,  as  no  ball  can 

What  is  the  geometrical  name  of  the  curve  in  which  a  ball 
moves  when  tossed  in  the  air  ?  Which  would  make  a  more 
perfect  parabola,  a  ball  of  lead  or  a  ball  of  cork  ?  (Of  lead, 
because  least  impeded  by  the  air.)     How  must  I  hold  a  plate 

10 


110  GEOMETRY. 

really  be  ;  the  table  perfectly  flat,  as  no  table  can 
really  be ;  the  blaze  of  the  lamp  a  single  bright 
point,  as  no  blaze  of  a  lamp  can  be.  It  is  easy  to 
imagine  exact  figures,  but  they  can  never  be  made. 
No  line  can  be  dra^yn  so  fine  and  true  that  a 
microscope  would  not  find  a  breadth  to  it,  or 
waving  irregularities  in  it. 

4.  The  parabola  is  a  very  useful  curve  ;  but  it 
would  be  difficult  to  explain  to  children  how  it  is 
used.  I  shall  tell  you  of  one  use,  before  the  end 
of  this  book. 

5.  On  a  smooth  board  draw  a  straight  line, 
such  as  B  c.     Near  the  middle  of  the  line,  as  at  a. 


drive  a  small  pin.  Put  one  edge  of  the  sqiiare 
card  against  the  pin,  and  one  corner  on  the  line 
B  c,  and  draw  a  pencil-line  along  the  edge  of  the 
card,  beginning  at  the  corner  on  b  c,  and  going  at 


so  that  the  edge  of  its  shadow  shall  be  a  parabola  ?  How  can 
I  draw  a  parabola  with  a  straight  edge  and  square  ?  What  is 
the  vertex  of  a  parabola  ?    How  near  the  vertex  does  the 


GEOMETRY. 


Ill 


right  angles  to  the  edge  that  is  against  the  pin  A. 
Do  this  with  the  card  in  a  great  many  different 
positions,  only  keeping  the  edge  against  the  pin 
and  the  corner  on  b  c,  and  you  will  make  a  place 
on  the  board  nearly  black  with  pencil-marks,  with 
a  curved  edge  on  the  inside,  around  A,  and  the 
curve  is  a  parabola. 

6.  The  point  A,  in  the  last  figure,  is  called  the 
focus  of  the  parabola.  The  point  in  the  parabola 
nearest  the  focus  is  called  the  vertex  of  the  para- 
bola.    The  line  b  c  is  a  tangent  at  the  vertex. 

7.  Let  c  M  be 
a  parabola,  and 
let  A  be  its  focus. 
Draw  D  E  paral- 

•  lei  to  the  tangent 
at  the  vertex,  and 
as  far  from  the 
vertex  as  the  ver- 
tex is  from  the  focus.  This  line  D  E  is  called  the 
directrix  of  the  parobola. 

8.  Any  point  in  the  parabola  is  just  as  far  from 
the  focus  as  from  the  directrix.  That  is  to  say, 
that,  if  we  take  any  point,  as  M,  and  draw  a  line 


directrix  pass  ?  In  what  direction  does  the  directrix  of  a  par- 
abola lie  ?  (Parallel  to  tangent  at  the  vertex.)  How  can  you 
describe  a  parabola  with  reference  to  its  focus  and  directrix  ? 
Let  the  teacher  copy  the  figure,  and,  drawing  a  tangent  at  the 


112  GEOMETRY. 

M  ?  at  right  angles  to  the  directrix,  and  also  a  line 
M  A,  the  two  lines  M  p  and  N  A  will  be  of  exactly 
equal  length. 

9.  A  parabola  may,  therefore,  be  described  as  a 
curve,  every  part  of  which  is  equally  distant  from 
a  point  called  the  focus,  and  from  a  straight  line 
called  the  directrix. 

10.  The  parabola  at  the  point  M  makes  exactly 
the  same  angle  with  the  line  M  P  that  it  does  w^ith 
the  line  ma. 

11.  If  H  c  is  the  radius  of  curvature  at  the 
point  c,  and  c  F  is  in  the  same  straight  line  with 
II  c,  then  c  F  is  just  half  as  long  as  H  c.  That  is 
to  say,  that  a  straight  line  drawn  at  right  angles 
to  any  point  in  a  parabola,  and  ending  in  the  direc- 
trix, is  just  half  as  long  as  the  radius  of  curvature  * 
at  that  point.  ' 


CHAPTER    XXXIII. 

THE    SHADOW    OF   A   BALL. 

1.  Any  curve  that  runs  round  into  itself  again 
encloses  an  oval.     The  word  oval  really  means 

point  M,  ask,  How  does  this  tangent  divide  the  angle  amp? 
How  can  you  tell  the  length  of  the  radius  of  curvature  at  any 
point  of  a  parabola  ? 

What  is  an  oval  ?    What  is  an  ellipse  ?    How  can  you  draw 


GEOMETRY.  113 

egg-shaped  ;  but,  in  geometry,  we    ^.-^ 

use  it  for  any  figure  bounded  by   f      ^^"^^^^ 

one  curve  line,  without  any  sharp    \^^^  J 

corner. 

2.  The  shadow  of  a  round  ball  falling  on  a  flat 
surface,  when  all  the  shadow  can  be  seen,  is  either 
a  circle  or  a  particular  kind  of  oval  called  an 
ellipse.  The  shadow  of  a  round  plate  is  also  an 
ellipse,  whenever  the  whole  shadow  can  be  seen  on 
one  flat  surface. 

3.  You  can  draw  an  ellipse  by 
driving  two  pins  into  a  board,  as 
at  A  and  B  in  the  figure,  and 
tying  a  string,  as  A  M  B,  one  end 
to  each  pin,  then  putting  a 
pencil-point,  as  at  M,  inside  the  string,  and  stretch- 
ing it  out,  and  moving  it  round. 

4.  The  points  where  the  pins  are  placed  are 
called  the  foci  of  the  ellipse.  Lines  to  the  foci 
from  any  point  in  the  ellipse,  as  at  M,  make  equal 
angles  v/ith  the  tangent  at  that  point. 

5.  The  nearer  the  foci  are  together,  using  the 
same  string,  A  M  B,  the  more  nearly  a  circle  does 


an  ellipse  with  a  string  and  two  pins  ?  What  is  the  name  of 
the  points  where  the  pins  are  ?  What  angles  do  the  two  parts 
of  the  string  make  with  that  part  of  the  ellipse  where  your 
pencil  is?  (Equal  angles.)  What  other  curve  is  the  end  of 
10* 


114  GEOMETRY. 

the  ellipse  become ;  so  that  if  the  foci  came  to- 
gether, the  ellipse  would  become  a  circle. 

6.  The  further  apart  the  foci  are  the  longer 
and  narrower  is  the  ellipse.  When  an  ellipse  is 
very  long,  and  very  narrow  in  proportion  to  its 
length,  each  end  of  the  ellipse  becomes  very  much 
like  a  parabola. 

7.  When  an  ellipse  is  very  exceedingly  long, 
the  ends  are  so  much  like  a  parabola  that  even 
geometers  call  them  parabolas.  We  call  the  path 
of  a  ball  tossed  in  the  air  a  parabola,  although  in 
reality  it  is  one  end  of  a  very  long  ellipse,  nearly 
four  thousand  miles  long,  with  one  focus  at  the 
centre  of  the  earth.  But  a  real  parabola  is  an 
dlipse  so  long  that  it  has  no  other  end  at  all :  it 
only  has  one  end  and  one  focus. 

8.  The  moon  goes  round  tte  earth  in  an  ellipse ; 
the  earth  goes  round  the  sun  in  an  ellipse.  And, 
if  you  were  to  cut  the  earth,  or  sun,  or  moon,  in 
two,  with  a  straight  cut,  the  cut  surface  would  be 
either  an  ellipse  or  a  circle,  according  to  the  direc- 
tion in  which  you  cut  it.  If  the  earth  is  cut  in 
two  from  east  to  west,  the  section  is  a  circle ;  if 


a  very  long  ellipse  like  ?  When  the  two  foci  of  an  ellipse  are 
brought  near  together,  what  curve  does  the  ellipse  become 
like  ?  Can  j^ou  explain  how  a  carpenter  draws  an  ellipse  by 
a  **  trammel  and  slots  "  ?    Did  you  ever  notice  an  elbow  in  a 


aEOMETRY. 


in  any  other  direction,  an  ellipse ;  for  the  earth 
is  not  perfectly  round. 

9.  Carpenters  sometimes  draw  ellipses  by  means 
of  a  board  with  two  narrow  slits  in  it  at  right 
angles  to  each  other.  They  have  a  ruler  with  two 
pins  in  it,  as  at  A  and  B, 
and  a  pencil  in  the  end, 
as  at  c ;  and,  by  moving  / 
one  pin  in  one  slit,  and  / 
the  other  pin  in  the  other  \ 
slit,  the  pencil  c  will 
move  in  an  ellipse. 

10.  If  you  cut  a  round  stick  off  slanting  with  a 
sharp  knife,  at  one  cut,  the  cut  end  will  be  an 
ellipse. 


CHAPTER    XXXIV, 

THE   SHADOW    OF   A   REEL. 

1.  If  a  reel  for  winding  thread, 
such  as  is  represented  in  the  fig- 
ure, be  held  steadily  in  such  a 
position  that  its  shadow  from  a  lamp 


stove-pipe  ?    What  is  the  shape  of  the  seam  around  the  elbow 
of  the  stove-pipe  ? 

Did  you  ever  see' a  **  swift"  for  winding  yarn?    Did  you 


116 


GEOMETRY. 


will  fall  on  a  flat  wall,  and  then 

set  to  revolving,  the  sides  of  the 

shadow  will  be  curved,  and  the 

curve  is  called  an  hyperbola. 

2.  Suppose  we  have 
strings  tied,  at  various 
places,  on  a  horizontal 
wire,  A  B,  and  all  drawn 
straight  through  one  point, 
c.  Cut  them  all  off  on  a 
line  parallel  to  A  b.     Let 

the  strings,  after  being  thus  trimmed,  hang  straight 

down,  and  the  ends  will  hang  in  a  curve,  as  shown 

in    this    figure,    and    q 


]sr 


that  curve  will  be  an 
hyperbola.  This  will 
also  be  true  if  the 
strings  are  cut  off 
exactly  at  the  point 
c. 

3.  If  a  rope  is  tied  to  a  fixed  point,  say  the 
hook  A,  and  passes  over  a  fixed  pulley,  as  B,  not 


ever  see  it  standing  steady  on  a  table,  and  spinning  round 
very  fast?  Did  j^ou  notice  that  the  sides  looked  curved? 
What  curve  was  it  ?  Suppose  a  row  of  palings  set  in  a  straight 
line,  the  middle  one  the  shortest,  and  the  others  just  long 
enough  to  reach  the  top  of  the  middle  one,  if  they  were  leaned 


GEOMETRY. 


117 


on  a  level  with  A,  then 
a  weight  on  a  mova- 
ble pulley,  c,  will 
move,  as  you  raise  it 
by  pulling  the  rope 
over  B,  in  an  hyper- 
bola. 

4.  If  we  draw  a 
circle  round  a  centre, 
B,  marking,  also,  some 

point  outside  the  circle,  as  at  A,  and  then  make  a 
dot  at  every  point  which  we  can  find  situated,  like 
M,  as  far  from  the  point  A  as  from  the  circumfer- 


\ 


ence  of  the  circle  round  b,  these  dots  Avill  all  be 
in  an  hyperbola. 


against  it  with  their  bottoms  standing  where  they  now  do  ; 
what  curve  would  the  top  of  such  a  row  of  palings  make  ? 
(Section  two.)  In  what  curve  does  a  movable  pulley  move, 
wlien  the  fixed  pulley  is  not  on  a  level  with  the  fixed  end  of 


118 


GEOMETRY. 


5.  You  see,  theiij  that  an  hyperbola  can  be 
fancied  as  a  parabola  with  a  circumference,  instead 
of  a  straight  line,  for  a  directrix. 

6.  In  the  last  figure  the  points  A  and  b  are 
called  the  foci  of  the  hyperbola.  The  curve  is,  at 
each  point,  as  far  from  one  focus  as  from  the  direc- 
trix circle  drawn  round  the  other.  That  is,  M  p  is 
of  the  same  length  as  M  A. 

7.   If  we  hang  light  threads 
to  each  link  of  a  hanging  chain, 

Q 


isr 


such  as  Q  N,  and  cut 

their  lower  ends  off 

on  a  level  line,  and 

then  stretch  the    chain  out  perfectly   level,   the 

lower  ends   of  the  thread  will  arch   up  into  an 

hyperbola. 

8.  If  a  round  ball  hangs  exactly  under  a  lamp, 
over  a  level  table,  its  shadow  on  the  table  will  be  a 
circle.     But  if  the  ball  is  moved  to  one  side,  the 


the  rope  ?  How  do  a  parabola  and  an  hyperbola  compare  with 
each  other  ?  How  many  foci  has  a  parabola  ?  An  hyperbola  ? 
Can  you  tell  how  to  make  an  hyperbola  by  a  chain  and 
threads  ?     Can  you  describe  how  the  shadow  of  a  ball,  or 


I. 


GBOMETRY.  119 


ow  becomes  an  ellipse.  Now  raise  the  ball 
slowly,  and  the  shadow  will  begin  to  move  away 
from  the  lamp.  But  one  edge  will  move  away 
much  faster  than  the  o*her,  so  that  the  ellipse  will 
grow  longer  and  longer.  And  if  we  imagine  the 
table  to  be  so  large  that  we  cannot  see  the  edges 
of  it,  then,  when  the  upper  edge  of  the  ball  is  just 
on  a  level  with  the  lamp,  the  ellipse  will  be  so 
long  that  it  will  have  no  other  end,  and  the  end 
nearest  the  lamp  will  be  a  parabola.  If  we  raise 
the  ball  higher,  the  parabola  becomes  an  hyper- 
bola. And  when  the  ball  is  raised  so  high  that 
its  under  side  is  as  high  as  the  lamp,  the  shadow 
will  not  touch  the  table  at  all.  The  parabola  and 
hyperbola  are  made  by  the  shadow  of  the  lower 
side  of  the  ball. 

9.  If  you  take  a  plate  instead  of  a  ball,  you  can 
make  all  the  shadows,  circle,  ellipse,  parabola,  and 
hyperbola,  by  a  little  pains-taking  to  hold  the 
plate  at  the  proper  angle  with  the  table  for  the 
circle  and  ellipse.  For  the  parabola  and  hyper- 
bola less  care  is  required  ;  only  that  tbx  a  parabola 
the  upper  edge  of  the  plate  must  be  just  as  high 
as  the  light ;  and  for  an  hyperbola  the  plate  must  be 


plate,  may  be  made  to  grow  from  a  circle  into  an  hyperbola  ? 
What  other  two  curves  does  it  become  before  becoming  an 
hyperbola  ? 


120  GEOMETRY. 

higher.     The   shadow  of  the   lower  edge  of  the 
plate  makes  the  parabola  or  hyperbola. 


CHAPTER    XXXV. 

THE    cow's  FOOT  IN   A   CUP   OF   MILK. 

1.  I  HAVE  already  told  you  how  to  make  the 
bright  curve  called  by  children  the  cow's  foot  in  a 
cup  of  milk.  I  have  also  told  you  how  to  draw  a 
parabola  by  drawing  lines  tangent  to  it  until  all 
the  paper  outside,  the  parabola  is  blackened  by 
pencil-marks.  I  have  also  told  you  how  to  draw 
a  hypocycloidj  by  putting  a  short  ruler  across  one 
corner  of  your  slate,  keeping  one  end  against  the 
frame  at  the  bottom,  and  the  other  end  against  the 
frame  at  the  side,  and  drawing  pencil-marks  the 
whole  length  of  the  ruler  on  the  side  next  the  cor- 
ner. The  corner  will  become,  by  making  the  marks 
at  a  great  many  different  angles  with  the  frame, 
whitened  with  pencil-marks,  all  tangent  to  an  arch 
of  a  hypocycloid. 

These  three  curves  are,  in  one  respect,  alike: 
they  are  made  by  drawing  tangents  to  them.     For 


How  are  curves  drawn  by  drawing  only  straight  lines? 
"What  is  the  geometrical  name  for  all  curves  made  by  reflected 


GEOMETRY.  121 

the  cow's  foot  is  made  by  bright  straight  lines  of 
reflected  light,  all  tangent  to  an  epicycloid. 

2.  And  whenever  light  is  reflected  from  the 
inside  of  a  polished  curve,  the  reflected  light 
makes  a  bright  curve  of  some  kmd,  just  as  the 
light  reflected  from  the  inside  of  a  circle  makes 
an  epicycloid. 

3.  Curves  made  by  reflected  light  arc  called 
caustics.  The  cow's  foot  in  a  cup  is  a  caustic 
made  by  a  circle.  The  caustic  made  by  a  circle  is 
an  epicycloid. 

4.  Suppose  you  had  a  table  so  arranged  that  the 
setting  sun  should  shine  over  its  surface.  If  on 
this  table  we  should  set  narrow  strips  of  tin  on 
their  edges,  they  would  reflect  the  sun-light  and 
make  bright  curves  or  caustics  on  the  table. 

5.  If  the  tin  were  bent  into  a  half  of  a  circle, 
the  caustic  made  by  it  would  be,  as  you  already 
know,  an  epicycloid  such  as  would  be  made  by 
one  circle  rolling  on  another  of  twice  its  diam- 
eter. 

6.  If  another  strip  were  bent  into  the  form  of  a 


liglit  ?  What  is  the  caustic  made  by  a  circle  ?  What  is  the 
child's  name  for  it  ?  How  could  we  arrange  a  table  and  make 
caustics  with  the  curves  ?  What  is  there  peculiar  about  the 
caustic  of  a  parabola  ?  Of  a  cycloid  ?  Do  you  remember 
wliat  parallel  lines  are?  Concentric  curves?  What  is  the 
11 


122  GEOMETRY. 

parabola,  and  turned  in  such  a  direction  that  the 
sun-light  fell  at  right  angles  to  the  directrix  of  the 
parabola,  then  the  caustic,  instead  of  being  a  curve, 
would  be  a  single  bright  point  at  the  focus  of  the 
parabola. 

7.  If  you  bend  another  strip  into  an  arch  of  a 
cycloid,  and  turn  the  straight  line  which  joins  the 

ends  of  the  arch  at 
right  angles  to  the  sun- 
light, the  caustic  will 
be  two  arches  of  a  cy- 
cloid of  just  half  the  size,  as  shown  in  the  figure. 

8.  If  the  cycloid  be  turned  round  at  right 
angles  to  its  last  position,  so  that  the  straight  line 
joining  the  ends  of  the  arch  shall  be  parallel  to 
the  sun-light,  the  caustic  will  not  be  a  cycloid,  but 

will  be  curve  con- 
centric with  a  cy- 
cloid of  half  the  size. 

-a 

'In  the  figure,  A  b 
represents  the  strip 
of  tin  in  the  form  of  a  cycloid ;  c  is  the  point  of 
the  caustic,  and  D  is  one  arch  of  the  half-size 
cycloid  with  which  the  caustic  is  concentric. 


caustic  of  a  cycloid  turned  endwise  to  the  light  ?  What  caus- 
tic is  always  formed  when  the  light  falls  perpendicular  to  any 
part  of  a  curve  ?    When  the  light  falls  parallel  to  it  ?    Let 


I 


GEOMETRY.  123 


9.  Whatever  curved  form  the  tin  may  be  bent 
into,  the  caustic  will  have  some  curious  properties, 
which  I  will  now  tell  you. 

Wherever  the  light  falls  at  right  angles  to  the 
curved  tin,  as  at  M.  the  caustic  will  be  at  the  mid- 
dle of  the  radius  of  curvature,  and 
the  radius  of  curvature  will  be  tan- 
gent to  the  caustic,  as  the  radius 
M  D  is  tangent  to  the  caustic  c,  just 
half  way  from  M  to  D.  And  the 
radius  of  curvature  of  the  caustic 
at  this  place,  c,  will  be  just  one  ^\M 
quarter  of  the  radius  of  curvature  of  the  evolute 
of  the  tin  at  M.  If  E  is  the  radius  of  curvature 
of  the  evolute,  then  R,  the  radius  of  curvature  of 
the  caustic,  will  be  parallel  to  E,  and  be  one  quar- 
ter as  long  as  E.  And  both  R  and  E  will  be  at 
right  angles  to  the  radius  M  D. 

But  wherever  the  sun-light  falls  as  a  tangent  to 
the  inside  of  the  curved  strip  of  tin,  the  caustic 
will  also  be  a  curve  tangent 
to  the  inside  of  the  tin,  and  '^//^ 
its.  radius  of  curvature  will 
be  exactly  three  quarters  ^ji 
the  radius  of  curvature  of 


the  teacher  copy  the  figures,  and  go  over  all  the  peculiar  points 
of  section  nine.     What  would  be  the  caustic  formed  by  a  lamp 


124  GEOMETRY. 

the  tin  curve.  Thus,  if  M  N  is  a  curve  on  the 
inside  of  vfhich  the  sun-light  falls  parallel  to  the 
curve  at  M,  then  M  o  will  be  the  caustic,  and  its 
radius,  M  Q,  will  be  exactly  three  quarters  of  M  p, 
the  radius  of  M  N. 

10.  All  through  this  chapter  on  caustics  I  have 
spoken  only  of  those  that  are  made  when  the  light 
is  at  a  great  distance  from  the  polished  tin.  If  we 
bring  a  lamp  near  the  polished  curve,  the  caustics 
made  by  this  lamp-light  w^ill  be  very  different. 

11.  A  lamp  placed  in  the  centre  of  a  circle 
would  not  make  a  curved  caustic,  but  all  the  light 
would  be  thrown  back  to  one  point,  in  the  centre, 
where  the  lamp  itself  stood. 

12.  A  lamp  placed  in  focus  of  an  ellipse  would 
not  make  a  curved  caustic,  but  all  the  light  would 
be  thrown  to  one  point,  the  other  focus  of  the 
ellipse. 

13.  A  lamp  placed  in  the  focus  of  a  parabola 
would  not  make  a  curved  caustic ;  but  all  the 
light  would  be  thrown  straight  out  in  parallel 
lines  perpendicular  to  the  directrix  of  the  parab- 
ola. For  this  reason  men  have  polished  reflect- 
ors in  the  shape  of  a  paraboloid  to  place  behind 


in  the  centre  of  a  circle  ?  In  the  focus  of  an  ellipse  ?  In  the 
focus  of  a  parabola  ?  Did  you  ever  notice  the  reflector  in  front 
of  a  locomotive  engine  ?    What  is  it  for  ? 


la] 

m 


GEOMETRY. 


125 


lamps  when  they  want  to  throw  out  the  light  in 

►ne  direction.     They  use   such  mirrors  in  light- 

ouseSj  to  throw  the  light  out  over  the  ocean ;  and 

ey  use  them  in  front  of  locomotive  engines,  to 

throw  light  straight  forward  on  the  track  by  night. 


CHAPTER    XXXVI. 

SOLID    GEOMETRY. 

1.  All  the  figures  which  I  have  told  you  about 
are  such  as  could  be  drawn  on  a  flat  sheet  of  paper. 
Before  I  finish  my  book  I  will  tell  you  a  little 
about  a  few  solid  bodies. 

2.  Cut  out  of  a 
stifi"  piece  of  paper 
six  equal  squares,  as 
I  have  drawn  them 
here ;  and  then  fold 
the  paper,  at  each  line  where  the  squares  join  each 
other,  to  a  right  angle.  You  will  thus  make  the 
six  equal  squares  shut  up  a  space  like  a  box. 

3.  This  solid  figure,  bounded  by  six  equal 
squares,  is  called  a  cube. 

What  is  the  name  of  a  solid  bounded  by  six  equal  squares  ? 
What  is  the  difference  between  a  yard  of  tape,  a  yard  of  oil- 
cloth, and  a  yard  of  earth  ?  —  I  mean  between  the  three  mean- 
IP 


126  GEOMETRY. 

4.  A  cube  is  taken  as  the  measure  of  all  solids 
and  fluids.  A  gallon,  for  instance,  is  two  hundred 
and  thirty-one  cubic  inches  ;  that  is  to  say,  by  a 
gallon  of  water  we  mean  water  enough  to  fill  two 
hundred  and  thirty-one  little  cubes,  whose  faces 
are  square  inches.  Or,  for  another  instance,  a 
cord  of  wood  is  one  hundred  and  twenty-eight 
cubic  feet ;  that  is,  wood  enough  to  make  a  pile  as 
large  as  one  hundred  and  twenty  boxes,  whose 
sides  are  each  a  square  foot. 

5.  You  remember  that  the  measure  of  a  square 
is  found  by  multiplying  the  length  of  a  side  by 
itself  The  measure  of  a  cube  is  found  by  multi- 
plying the  length  of  a  side  twice  by  itself  If  I 
build  a  cube  with  a  side  of  three  inches,  one  face 
will  have  nine  square  inches  in  it,  and  the  cube 
will  be  made  up  of  three  layers,  each  with  nine 
cubic  inches  in  it.  So  that  the  cube  of  three 
inches  is  three  times  three  times  three  inches ; 
that  is,  twenty-seven  inches.  A  cube  of  four 
inches  would  have  sixteen  square  inches  on  a 
side,  and  consist  of  four  layers  of  sixteen  cubic 
inches  each  ;  that  is,  the  cube  of  four  inches  is 
sixty-four  inches. 


1 


ings  of  the  word  yard  in  those  phrases  ?  Do  you  know  how 
long  a  yard  is  ?  Did  you  ever  see  a  man  two  yards  high  ? 
Build  an  earthen  pyramid  as  tall  as  such  a  man's  head,  and 


GEOMETRY.  127 

6.  Similar  surfaces,  you  remember,  are  in 
proportion  to  the  cubes  on  corresponding  lines. 
Similar  solids  are  in  proportion  to  the  cubes  on 
corresponding  lines. 

7.  And  as  you  can  find  the  proportion  between 
tAYO  similar  surfaces  by  multiplying  the  number 
that  expresses  the  proportion  between  the  lines 
by  itself,  so  you  can  find  the  proportion  between 
the  solids  by  multiplying  the  number  that  ex- 
presses the  proportion  between  the  lines  twice  by 
itself. 

8.  Let  us  suppose,  for  instance,  a  heap  of  earth 
six  feet  high,  built  in  the  shape  of  the  great  pyra- 
mid in  Egypt,  which  is  six  hundred  feet  high. 
The  pyramid  would  be  one  hundred  times  as  high 
as  the  heap  of  earth ;  and,  being  of  the  same 
shape,  would  also  be  one  hundred  times  as  wide  at 
the  bottom.  But  the  pyramid  would  cover  a  hun- 
dred hundred,  that  is,  ten  thousand  times  as  much 
land  as  the  heap  of  earth  :  and  it  would  be  a  liun- 
dred  times  ten  thousand,  that  is  to  say,  one  million 
times,  as  large  as  the  heap  of  earth. 


how  much  higher  -would  the  great  pyramid  of  Egypt  be  ?  How 
much  hirger  ?  AYhat  is  the  highest  hill  in  this  neighborhood? 
How  high  is  it  ?  How  many  such  hills  set  one  on  top  the  other 
would  it  take  to  be  as  high  as  the  White  Hills  ?  Suppose  Mount 
Washington  were  of  the  same  shape  as hill,  how  many 


128  GEOMETRY. 

9.  The  highest  mountains  in  the  United  States, 
east  of  the  Mississippi  river,  are  about  ten  times 
as  high  as  the  great  pyramid ;  and  if  one  of  them, 
one  of  the  White  Hills,  for  example,  were  cut  into 
the  same  shape  as  the  great  pyramid,  it  would 
cover  one  hundred  times  as  much  land,  and  have 
one  thousand  times  as  much  stone  in  it,  as  one  of 
the  pyramids. 

10.  The  highest  mountains  of  Thibet  are  nearly 
five  times  as  high  as  the  White  Hills  of  New 
Hampshire.  If  one  of  the  highest  mountains  of 
Thibet  were  of  the  same  shape  as  one  of  the  White 
Hills,  it  would  have  twenty-five  times  as  much 
land  on  its  sides,  and  would  cover  twenty-five 
times  as  much  space.  And  it  would  take  one 
hundred  and  twenty-five  mountains  like  the  White 
Hills  to  make  one  of  the  Himalaya  mountains. 

11.  I  wish  you  to  remember  very  carefully  that 
when  two  things  are  of  the  same  shape,  all  cor- 
responding lines  are  in  the  same  proportion  to 
each  other  ;  all  corresponding  surfaces  are  in  pro- 
portion to  the  squares  on  those  lines  ;  and  all  cor- 
responding solids  in  proportion  to  the  cubes  on 
those  lines. 


such  hills  would  it  take  to  build  Mount  Washington  ?  Sup- 
pose Gulliver  to  be  twelve  times  as  high  as  the  Lilliputs,  but 
shaped  just,  like  them,  how  many  times  larger  would  his  nose 


GEOMETRY.  129 

12.  I  wish  you  also  to  remember  that,  if  the 
number  expressing  the  proportion  between  the  lines 
of  two  similar  solids  be  multiplied  by  itself,  the 
product  will  express  the  proportion  between  the  cor- 
responding surfaces ;  and,  if  it  be  again  multiplied 
by  itself,  the  product  will  express  the  proportion 
between  the  corresponding  solidities.  If  the  diam- 
eter of  a  foot-ball  be  five  times  that  of  a  batting- 
ball,  the  surface  of  the  foot-ball  will  be  twenty- 
five  times  as  much,  and  the  size  of  the  foot-ball 
will  be  one  hundred  and  twenty-five  times  as  much, 
as  that  of  the  other  ball. 


CHAPTER    XXXVII. 

CONIC   SECTIONS.    • 

1.  You  will  very  often  hear  or  read  about  conic 
sections  ;  men  began  to  study  them  more  than  two 
thousand  years  ago,  and  have  not  yet  learned  all  the 
useful  things  that  can  be  known  about  them.  By 
conic  sections,  we  mean  the  circle,  the  ellipse,  the 
parabola,  and  the  hyperbola.  I  have  told  you  a 
little  about  these  curves  ;  enough,  I  hope,  to  make 

be  than  theirs  ?  How  many  times  larger  would  his.  thumb- 
nail be  ?  How  many  times  larger  would  his  finger  be  than 
theirs  ? 


130 


aEOMETKY. 


you  want  to  learn  more  ;  and  I  will^  in  this  chap- 
ter, tell  you  why  they  are  called  conic  sections. 

2.  Cut  out  of  pasteboard  a  figure  bounded  by 

an  arc  and  two  radii, 
such  as  A  B  c.    Curve 

\^  it  up  equally,  and  join 
the    edge  A  c  to  the 

enclose  (on  all  sides  but  one)  a  space;  and  the 
figure  thus  formed  is  called  a  right  cone. 

3.  Set  a  right  cone  up  upon  a  plane,  and  the 
arc  A  B  will  become  a  circle,  such  as  d  e  in  the 

figure.  If  we  fancy  a 
post  standing  straight  up 
in  the  centre  of  a  cir- 
cle, and  a  longer  straight 
pole  tied  by  one  end 
XJB  to  the  top  of  the  post, 

while  the  other  end  just  reaches  the  circumference 
of  the  circle  ;  if  we  then  fancy  this  lower  end  car- 
ried around  the  circumference,  the  pole  will  mark 
out  in  the  air  the  surface  of  a  right  cone. 

4.  Cut  a  right  cone  in  two  by  a  plane  parallel 


If  a  right  triangle  could  spin  round  on  one  leg,  the  other 
leg  will  describe  a  circle,  and  the  hypothenuse  will  go  round  a 
solid  body  in  the  air  ;  now  what  is  its  geometi^cal  name  ? 
How  shall  I  cut  a  right  cone  so  as  to  make  the  cut  surface  a 


I 


GEOMETRY. 


131 


AJB 


to  the  plane  on  which  it  sits,  and  the  cut  surface 
will  be  a  circle. 

5.  Cut  a  right  cone 
in  two  by  a  plane  in- 
clined to  the  plane  on 
which  it  stands,  and  the 
cut  surface  will  be  an 
ellipse. 

^^  Section"  means  a  cut  surface,  and  ^^ conic" 
means  belonging  to  a  cone ;  so  that  you  can  now 
understand  why  these  curves  are  called  ^' conic 
sections  "  ;  it  is  because  they  can  be  made  by  cut- 
ting a  cone. 

6.  Cut   the    cone  by 

a  plane  parallel  to  one 

side  of  the  cone,  and  the 

cut    surface   will    be  a 

A-B  parabola. 

7.  Cut  the  cone  by  a  plane 
making  a  smaller  angle  with  the 
centre-post  than  the  sides  do,  and 
the  cut  surface  will  be  an  hy- 
perbola. 

8.  Now,  if  you  will  turn  back  and  read  chap- 
ter  XXXIV.,  section   eight,   again  carefully,  you 


AB 


circle  ?    An  ellipse  ?    A  parabola  ?  An  hyperbola  ?    How  can 
you  make  part  of  a  cone  in  the  air  with  a  lamp  and  ball  ? 


132  GEOMETRY. 

will  see  that  the  shadow  of  a  ball  is  part  of  a 
cone  in  the  air,  with  the  vertex  or  point  of  the 
cone  in  the  blaze  of  the  lamp,  and  that  the  flat 
table  is  a  plane  that  makes  conic  sections  of  the 
shadow. 

9.  If  you  were  to  bend 
the  pasteboard  cone,  so 
as  to  make  d  e  some 
other  shape  than  a  cir- 
cle, the  cone  would  no 
longer  bo  a  right  cone. 
A  right  cone  has  a  circle  for  its  base,  and  the 
vertex  of  the  cone  is  directly  over  the  centre  of 
the  base. 


A3 


CHAPTER    XXXVIII. 

THE   SPHERE. 

1.  If  a  circle  should  spin  round  on  one  of  its 
diameters,  the  circumference  would  enclose  a  space 
called  a  sphere. 

2.  Solid  bodies  in  the  shape  of  a  sphere  are 
called  balls  or  globes.     The  marbles  with  which 

How  can  you  show  sections  of  such  a  cone  ?    What  is  a  right 
cone  ? 

What  is  the  geometrical  name  for  a  perfectly  round  solid  ? 
What  do  we  call  a  solid  body  that  has  the  form  of  the  georaet- 


GEOMETRY.  133 

boys  play,  are  usually  very 
perfect  globes.  A  soap- 
bubble  blown  thin,  and  free 
from  any  hanging  drop  of 
suds,  floating  in  still  air,  is 
a  very  perfect  sphere. 

3.  Cut  a  sphere  by  any  plane,  and 
the  cut  surface  will  be  a  circle. 

4.  When  the  plane  goes  directly 
through  the  centre  of  the  sphere,  the 

circle  thus  made  is  called  a  great  circle  of  the 
sphere.  All  great  circles  in  a  sphere  are  of  the 
same  size  as  the  circle  which  we  imagined  spinning 
to  create  the  sphere. 

5.  The  diameter  of  the  great  circle  is  called  the 
diameter  of  the  sphere,  and  the  radius  of  the  great 
circle  is  called  the  radius  of  the  sphere. 

6.  The  surface  of  the  sphere  is  exactly  four 
times  the  surface  of  a  great  circle.  A  ball  three 
inches  in  diameter  would  take  as  much  leather  to 
cover  it  as  would  make  four  circles,  each  three 
inches  in  diameter;  or  one  circle  six  inches  in 
diameter. 


rical  solid  ?  What  examples  of  balls  or  globes  can  you  give 
me  ?  How  must  a  circle  move  to  have  it  describe  a  sphere  ? 
What  shape  is  the  section  of  a  sphere  by  a  plane  ?  What  is 
a  great  circle  on  a  sphere  ?  How  large  is  the  surface  of  a  sphere  ? 

12 


134  GEOMETRY. 

7.  You  remember  that  the  measure  of  a  circle 
is  found  by  multiplying  the  square  of  its  diameter 
by  one  quarter  of  n.  As  the  surface  of  the  sphere 
is  four  times  as  great,  it  is  found  by  multiplying 
the  square  of  its  diameter  by  re  itself.  The  surface 
of  a  ball  three  inches  in  diameter,  for  instance, 
will  be  nine  square  inches  multiplied  by  7t, 

8.  The  solid  measure  of  a  sphere  is  found  by 
multiplying  the  cube  of  the  diameter  by  one  sixth 
of  TT.  The  cube  of  the  diameter  will  just  enclose 
the  sphere,  and  each  of  the  six  sides  of  the  cube 
will  be  a  tangent  plane  to  the  sphere. 

9.  For  roughly  judging  of  circles  and  spheres 
we  call  n  about  three.  That  is  to  say,  a  circum- 
ference is  a  little  more  than  three  times  the  diam- 
eter ;  a  circle  is  a  little  more  than  three  fourths  of 
the  square  on  the  diameter ;  and  a  sphere  is  a  little 
more  than  half  the  cube  on  the  diameter. 

10.  To  be  more  exact,  we  call  ti  twenty- two 
sevenths.  That  is  to  say,  a  circumference  is 
twenty-two  sevenths  of  the  diameter ;  a  circle, 
eleven  fourteenths  of  the  square  on  the  diameter  ; 
and  a  sphere,  eleven  twenty-firsts  of  the  cube  on 
the  diameter. 

How  large  is  the  solidity  of  a  sphere  ?  What  is  the  largest 
body  of  all  having  the  same  surface  ?  Into  what  shape  must 
I  pack  anything  to  make  it  expose  least  surface  ?  If  you  have 
studied  any  arithmetic,  you  may  now  tell  me  how  to  calculate 


GEOMETRY.  135 

11.  To  be  still  more  exact,  we  call  tt^  8-1416. 
That  is  to  say,  to  find  the  circumference,  multiply 
a  diameter  by  3*1416  ;  to  find  the  size  of  the  cir- 
cle, multiply  the  diameter  by  itself,  and  then  by 
•7854  ;  and  to  find  the  contents  of  a  sphere,  multi- 
ply the  diameter  twice  by  itself,  and  then  by 
•5236. 

12.  As  the  circle  is  the  largest  of  isoperimetri- 
cal  figures,  so  the  sphere  is  the  largest  of  all  bodies 
having  the  same  amount  of  surface.  If  you  roll  a 
piece  of  putty  into  a  round  ball,  it  will  haTe  less 
surface  than  it  could  have  in  any  other  form. 

13.  But  I  think  I  have  made  my  book  long 
enough.  I  hope  you  have  liked  it,  and  I  hope 
that  at  some  time  you  will  study  more  Geometry, 
and  learn  how  to  prove  the  truth  of  all  I  have  told 
you.  You  will  then  find  that  there  is  a  great  deal 
to  be  learned  about  what  men  already  know  of 
Geometry,  and  that  there  is  a  great  deal  that  is 

'  not  known,  at  least  by  any  man.  Of  course,  the 
great  Creator,  who  has  made  all  things  in  number, 
w^eight  and  measure,  knows  everything.  And  the 
more  we  know,  the  more  clearly  we  shall  see  how 
greiit  is  His  knowledge,  how  wonderful  his  wisdom, 

roughly  the  circumference,  the  surface  of  a  circle,  the  surface 
of  a  sphere,  and  the  solidity  of  a  sphere,  when  you  know  the 
diameter.  How  shall  we  calculate  the  same  more  exactly? 
How  still  more  exactly?    Is  there  anymore  Geometry  to  be 


186  GEOMETEY. 

and  how  beautiful  the  mannei"  in  which  he  has  used 
what  we  call  Geometry  in  the  forms  he  has  given 
to  all  things  on  the  earth  or  in  the  sky. 


learned  than  what  is  taught  in  this  book  ?  Are  there  any  new 
things  yet  to  be  discovered  in  Geometry  ?  Should  you  like  to 
learn  more  about  it  when  you  grow  older  ? 


THE   END. 


PRACTICAL   QUESTIONS  AND   PROBLEMS  FOR 
REVIEW  BY  THE  OLDER  SCHOLARS. 


The  chapters  referred  to  may  not  always  furnish  a  direct 
answer  or  solution  ;  but  they  will  always  suggest  the  true 
solution,  which  is  not  always  to  be  reasoned  out,  but  is  to  be 
seen  by  the  mind's  eye.  Similar  questions  can  be  multiplied 
indefinitely  by  a  skilful  teacher. 


Chapter  hi.  —  What  is  the  best  way  of  making  a  garden- 
path  straight  ?  How  does  a  carpenter  mark  a  long  straight 
line?  How  will  you  make  a  short  straight  line  on  paper? 
The  railroad  from  my  house  to  Boston  is  about  nine  miles 
long,  the  carriage-road  about  eight  ;  which  is  more  nearly 
straight?  In  travelling  the  carriage-road  to  Boston  I  cross 
the  Fitchburg  railroad  only  once.  I  am  on  the  north  side  of  it 
at  starting  ;  when  I  get  into  Boston ,  on  which  side  of  me  must 
I  look  for  the  depot?  But  I  cross  the  Worcester  railroad 
twice  ;  on  which  side  of  me  shall  I  look  for  the  depot  of  that 
road  ?  If  Boston  lies  exactly  east  of  my  house,  how  can  I 
manage  to  drive  my  horse  there  without  having  his  head  once 
turned  exactly  to  the  east  ?  Can  I  do  it  without  having  either 
head  or  tail  turned  to  the  east  ? 

12^ 


138  QUESTIONS   AND    PROBLEMS. 

Chaptee  IV.  —  Suppose  a  straight  stick  is  made  to  turn 
upon  a  pin  thrust  through  the  middle  of  it,  which  end  will 
move  the  faster  ?  (Neither.)  Which  end  will  alter  its  direc- 
tion most  rapidly  ?  If  the  pin  is  thrust  into  a  straight  line 
that  will  not  move,  such  as  a  crack  in  the  floor,  which  end  of 
the  stick  will  make  the  larger  angle  with  the  crack  ? 

Chapter  v.  —  If  I  hang  tw6  plumb-lines  from  the  ceiling, 
from  nails  that  are  just  one  foot  apart,  how  far  apart  will  the 
lines  be  six  feet  below  the  ceiling  ? 

Chapter  vi.  —  In  a  triangle  whose  sides  are  three,  four  and 
five  inches,  which  is  the  largest  and  which  the  smallest  angle  ? 
If  a  leaning  pole  makes  an  angle  equal  to  one  third  of  a  right 
angle,  with  a  plumb-line,  what  angle  does  it  make  with  a  level 
line  passing  through  the  foot  of  the  pole  and  under  the  bob 
hung  from  its  summit?  What  angle  will  it  make  with  any 
other  level  line  passing  through  its  foot  ?  What  angle  will  the 
pole  make  with  a  level  line  passing  through  its  foot,  at  right 
angles  to  one  passing  from  the  foot  under  the  bob  ? 

Chapters  vii.,  viii.,  ix.  —  Suppose  that  I  set  a  stake  seven 
feet  high  in  a  level  piece  of  ground,  and  measure  its  shadow 
and  the  shadow  of  other  things  on  level  ground  as  follows  : 
When  the  shadow  of  the  stake  was  ten  feet,  that  of  the  house 
was  thirty  feet  ;  when  that  of  the  stake  was  nine  feet,  that  of 
my  poplar-tree  was  forty-one  ;  when  that  of  the  stake  was 
eight,  that  of  my  cherry-tree  was  twenty-six  ;  when  that  of 
the  stake  was  six,  that  of  the  church-steeple  was  one  hundred 
and  twenty-one.  What  is  the  height  of  the  steeple,  poplar- 
tree,  cherry-tree,  and  house  ? 

Chapter  x.  —  I  have  one  post  and  two  rails  to  make  a 


QUESTIONS   AND    PROBLEMS.  139 

fence  to  keep  the  cattle  from  a  young  tree  that  has  sprung  up 
by  the  fence  in  my  pasture  ;  how  shall  I  make  the  largest  pen 
for  it  ?  Another  tree  stands  in  the  middle  of  the  field  ;  my 
only  materials,  for  making  a  defence  around  it,  are  three 
posts,  one  raU,  and  a  piece  of  rope  longer  than  the  rail.  In 
what  is  the  largest  triangle  I  can  make  ? 

Chapter  xi.  —  How  large  is  each  angle  in  an  isosceles  right 
triangle  ?  How  many  such  triangles  will  it  take  to  make  a 
square  ?  What  does  a  carpenter  mean  by  a  mitre-joint  ?  Do 
you  know  how  a  carpenter  makes  a  mitre-joint  ? 

Chapter  xii.  —  If,  in  a  field  with  four  straight  sides,  we 
find  all  the  sides  of  the  same  length,  what  may  we  know  about 
the  angles  ?  If,  in  such  a  field,  we  find  the  sides  all  equal,  and 
two  of  the  adjacent  angles  equal,  how  large  is  each  angle  in 
the  field  ?  and  what  do  you  call  the  shape  of  the  field  ?  I  have 
seen  braces  that  were  of  no  use.  What  is  the  proper  way  to 
make  them  ?  There  is  another  use  of  three  points  not  exactly 
like  this.     Why  is  a  three-footed  table  sure  to  stand  steady  ? 

Chapters  xiii.  and  xiv.  —  How  many  yards  of  painting  on 
the  side  of  a  house  forty-two  feet  long  and  twenty-one  feet 
high  ?  How  many  square  inches  in  a  pane  of  seven  by  nine  ? 
How  many  in  a  pane  of  eight  by  ten  ?  How  many  feet  of  land  in 
a  lot  with  two  sides  of  eighty  feet  each  and  two  of  thirty-nine 
feet  each,  if  the  angles  are  such  that  the  eighty  feet  sides  are 
only  thirty-five  feet  apart  ?  How  do  you  know  that  this  lot  is 
a  parallelogram  ? 

Chapter  xv.  —  How  many  feet  of  land  in  a  triangle  whose 
sides  are  thirty,  forty  and  fifty  feet  ?  How  many  in  a  triang]|! 
whose  sides  are  twelve,  five  and  thirteen  feet  ?    How  do  you 


140  QUESTIONS  'and    PROBLEMS. 

know  that  these  triangles  are  right  triangles  ?  Suppose  that  a 
quadrangle  has  sides  of  three,  four,  twelve  and  thirteen  feet, 
and  that  the  sides  of  three  and  four  feet  join  in  a  square  cor- 
ner, how  many  feet  of  land  does  it  include  ?  How  do  you 
know  that  this  quadrangle  can  be  divided  into  two  right 
triangles  ? 

Chapters  xvi.  to  xxi.  —  If  I  have  a  piece  of  the  felloes  of  a 
wheel  and  want  to  find  out  how  large  the  whole  wheel  is,  what 
shall  I  do  ?  How  would  you  lay  out  a  garden-path  in  a  circle  ? 
How  would  you  make  two  paths  at  right  angles  to  each  other  ? 
How  will  you  make  a  circle  on  the  blackboard  ?  How  will  you 
make  two  paths  run  at  an  angle  of  sixty  degrees  with  each 
other  ?  If  a  steamboat's  tiller  is  lashed  fast  in  any  position, 
in  what  curve  will  the  boat  run  ?  How  will  you  make  the 
circle  larger  ?  If  there  is  a  circle  drawn  on  the  blackboard, 
how  can  I  draw  a  tangent  to  it  at  any  particular  spot  in  the 
circumference  ?  There  are  four  ways,  one  from  xx.  7,  one  from 
XX.  8,  one  from  xxi.  5,  and  one  from  xix.  7.  These  ways  have 
their  special  advantages  and  disadvantages.    Point  them  out. 

Chapters  xxii.  and  xxiii.  —  How  shall  I  find  a  point  at  an 
equal  distance  from  three  given  points  ?  How  can  you  find 
the  place  that  is  equally  distant  from  three  of  the  corners  of 
this  room  ?  How  can  you  find  a  spot  equally  distant  from  the 
east  side,  the  south  side,  and  the  diagonal  of  the  room  that 
runs  south-west  ?  What  three  kinds  of  polygons  with  equal 
sides  and  equal  angles  can  be  laid  together  like  bricks  in  a 
pavement,  and  fill  up  all  the  space  ? 

Chapters  xxiv.  and  xxv.  —  Suppose  the  earth  to  be  eight 
thousand  miles  in  diameter,  what  is  its  circumference  ?    "What 


QUESTIONS   AND    PROBLEMS.  141 

is  the  length,  of  an  arc  of  seventy  degrees  in  a  circle  of  one 
foot  radius  ?  Which  weighs  most,  a  square  sheet  of  tin  eleven 
inches  on  a  side,  or  a  round  piece  of  the  same  thickness  thir- 
teen inches  in  diameter  ?  If  a  sheet  of  tin  four  inches  square 
weighs  an  ounce,  what  will  a  sheet  a  foot  square  weigh  ?  What 
will  a  circle  ten  inches  in  diameter  weigh  ?  What  is  the  differ- 
ence between  a  piece  of  land  four  rods  square,  and  a  piece  of 
four  square  rods  ?  What  is  the  difference  between  a  foot  square 
and  a  square  foot?  If  a  church-spire  is  one  hundred  and 
twenty  feet  high,  how  much  more  paint  will  it  take  to  paint 
the  church  than  to  paint  a  model  of  it,  with  a  spire  twelve 
inches  high  ?  Is  a  square  foot  of  sheet-lead  necessarily  in  a 
square  form  ?  What  proportion  in  the  cloth  required  to  clothe 
a  man  five  feet  high,  a  man  five  feet  ten  inches,  and  a  man  six 
feet,  supposing  the  three  men  to  be  of  the  same  form,  and 
dressed  in  the  same  fashion  ? 

Chapters  xxvi.,  xxvii.,  xxviii.  —  How  far  does  the  head 
of  a  spike  in  the  tire  of  a  wheel  four  feet  in  diameter,  travel 
while  the  wagon  goes  four  miles  on  a  level  road  ?  What  is  the 
radius  of  curvature  of  its  path  when  it  is  at  the  top  of  the 
wheel  ?  When  it  is  two  feet  from  the  ground  ?  When  it  is 
one  foot  from  the  ground  ?  A  pendulum-bob  swings  in  an  arc 
of  a  circle  ;  how,  from  xxviii.  4,  can  you  devise  a  plan  to  make 
it  swing  in  the  arc  of  a  cycloid  ?  What  is  the  shortest  path 
from  one  point  to  another  ?  Is  the  shortest  path  always  quick- 
est ?  When  is  it  not,  for  whom  or  what  is  it  not,  and  why 
not? 

Chapters  xxix.  and  xxx.  —  In  a  machine  called  a  photom- 
eter, a  bead  is  placed  on  the  rim  of  a  wheel  rolling  inside  of  a 


142  QUESTIONS   AND    PROBLEMS. 

hoop  of  just  double  the  diameter  ;  in  what  path  does  the  bead 
move  ?  In  a  railroad  curve  the  cars  cannot  turn  if  the  radius 
is  too  small  ;  what  objection  to  joining  two  straight  tracks 
which  are  at  right  angles  to  each  other  by  a  curved  track 
marked  out  by  drawing  many  lines  of  equal  length  across  the 
corner  ? 

Chapter  xxxi.  —  If  a  rope  weighs  one  pound  for  each 
yard,  and  I  tie  one  end  of  it  to  a  staple  in  the  wall,  how  hard 
must  I  -pull  horizontally  in  order  to  make  the  radius  of  curva- 
ture ten  feet  at  the  lowest  point  of  the  rope  ?  How  hard  to 
make  the  radius  of  curvature  twenty-one  feet?  One  hundred 
and  eight  feet  ?  What  is  the  radius  of  curvature  of  a  straight 
line  ?  How  hard  must  I  pull  horizontally  to  make  the  rope 
straight  ?  If  a  chain,  weighing  two  pounds  to  the  yard,  hangs 
between  two  posts  of  the  same  height,  and  the  curvature  of 
the  chain  in  the  middle  has  a  radius  of  five  feet,  with  what 
force  does  it  draw  in  each  post  ?  What  if  the  radius  is  thirty 
feet  ?  If  a  piece  of  thread  weighs  at  the  rate  of  an  ounce  to  a 
thousand  feet,  what  horizontal  force  is  required  to  make  the 
radius  of  curvature  a  mile  long  ?  But  what  to  draw  the  thread 
straight  ?  In  answering  any  of  these  questions  on  chapter 
XXXI.,  does  it  make  any  difference  how  long,  or  how  short  the 
rope,  chain  or  thread,  is  ? 

Chapters  xxxii.,  xxxiii.,  xxxi  v.  —  What  is  the  path  of  a 
rifle-ball  in  the  air  ?  Can  it  then  ever  go  straight  to  its  mark  ? 
Can  you  fancy  the  shape  of  the  evolute  of  an  ellipse  ?  If  a 
hypocycloid  of  four  arches  be  used  as  an  evolute,  but  the 
string  is  taken  on  two  opposite  sides  long  enough  to  wrap 
round  the  whole  arch,  and  on  the  other  sides  of  no  length. 


QUESTIONS   AND    PROBLEMS.  143 

what  sort  of  a  curve  would  it  produce  ?  (An  oval,  but  not  an 
ellipse. ) 

Chapter  xxxv.  —  What  shape  must  a  mirror  be  to  act  as  a 
burning  mirror  by  bringing  the  sun-light  to  a  point  ?  If  a 
piece  of  a  hollow  sphere  is  used,  at  what  distance  from  it  will 
the  imperfect  point  of  light  be  formed  ?  If  you  stand  in  the 
centre  of  a  field  bounded  by  a  circular  fence,  where  will  the 
echo  of  your  voice  sound  loudest  ?  If  the  walls  of  a  room  are 
in  the  form  of  an  ellinse,  and  a  man  stands  in  one  focus  and 
speaks,  where  will  the  echo  sound  loudest  ?  If  a  paraboloid 
reflector  were  placed  behind  the  whistle  of  a  locomotive,  what 
effect  would  it  have  on  the  sound  ? 

Chapter  xxxvi.  —  A  man  is  said  to  have  borrowed  a  heap 
of  peat-mud,  which  was  stacked  in  a  cubical  form,  four  feet 
on  a  side,  and  to  have  returned  two  heaps,  each  a  cube  of  three 
feet  on  a  side.  Did  he  make  a  just  return  ?  What  is  the  pro- 
portion between  the  length  of  a  hogshead  holding  one  hundred 
and  twenty-five  gallons,  and  a  keg  holding  one  gallon,  if  they 
are  of  the  same  shape  ?  If  the  smallest  of  the  three  men 
mentioned  on  page  141,  weighs  one  hundred  and  fifty  pounds, 
what  do  the  others  weigh  ?  If  a  man  five  and  a  half  feet  high 
weighs  one  hundred  and  sixty  pounds,  and  a  man  three  inches 
taller  weighs  one  hundred  and  eighty,  which  is  stouter  in  pro- 
portion to  their  height  ? 

C  APTER  XXXVII.  —  Suppose  a  pole  fastened  at  the  end  of 
a  horizontal  revolving  arm.  If  the  pole  lies  horizontal,  it  keeps 
in  a  horizontal  plane  ;  if  it  is  vertical,  it  describes  the  surface 
of  a  vertical  cylinder.  But  if  it  inclines  towards  the  centre- 
post  about  wliich  the  arm  revolves  ?    If  it  inclines,  but  not 


144  QUESTIONS   AND    PROBLEMS. 

directly  toward  the  centre-post?  Cut  such  a  hyperboloid 
surface  by  a  horizontal  j)lane,  and  what  will  the  section  be  ? 
Cut  it  by  a  vertical  plane,  what  will  the  section  be  ?  What 
chang'^.  as  you  move  the  vertical  plane  to  and  from  the  centre 
of  the  figure  ? 

Chapter  xxxviii.  —  How  many  cubic  feet  of  gas  will  fill  a 
round  balloon  seven  yards  in  diameter  ?  How  many  yards  of 
silk  three  quarters  of  a  yard  wide  will  it  take  to  make  such  a 
balk  on  ?  If  the  earth  were  eight  thousand  miles  in  diameter, 
and  a  perfect  sphere,  what  would  be  the  number  of  square 
miles  of  its  surface  ?  Of  solid  miles  in  its  contents  ?  To  how 
many  balls  thirteen  inches  in  diameter  would  it  be  equivalent  ? 
How  many  tons  would  it  weigh,  if  it  were  all  water,  one 
thousand  ounces  to  a  cubic  foot  ?  How  much  if  three  and  one 
half  times  that  weight  ? 


^o^ 


U.C.  BERKELEY  LlBRftRIES 


8327(57 


THE  UNIVERSITY  OF  CALIFORNIA  UBRARY 


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V  !Eiuer8on*is  Watts  ou  the  Mind* 

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V  RUSSELL'S  ENUNCIATION. 

5>  

^      Abbott's  L.ttt.e  Philosopheu.     Boss'JBt's  French  Woud  and  ' 
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